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Proofgold Proof
pf
Apply df_cart__df_img__df_domain__df_range__df_cup__df_cap__df_restrict__df_succf__df_apply__df_funpart__df_fullfun__df_ub__df_lb__df_altop__df_altxp__df_ofs__df_transport__df_colinear with
wceq
cofs
(
copab
(
λ x0 x1 .
wrex
(
λ x2 .
wrex
(
λ x3 .
wrex
(
λ x4 .
wrex
(
λ x5 .
wrex
(
λ x6 .
wrex
(
λ x7 .
wrex
(
λ x8 .
wrex
(
λ x9 .
wrex
(
λ x10 .
w3a
(
wceq
(
cv
x0
)
(
cop
(
cop
(
cv
x3
)
(
cv
x4
)
)
(
cop
(
cv
x5
)
(
cv
x6
)
)
)
)
(
wceq
(
cv
x1
)
(
cop
(
cop
(
cv
x7
)
(
cv
x8
)
)
(
cop
(
cv
x9
)
(
cv
x10
)
)
)
)
(
w3a
(
wa
(
wbr
(
cv
x4
)
(
cop
(
cv
x3
)
(
cv
x5
)
)
cbtwn
)
(
wbr
(
cv
x8
)
(
cop
(
cv
x7
)
(
cv
x9
)
)
cbtwn
)
)
(
wa
(
wbr
(
cop
(
cv
x3
)
(
cv
x4
)
)
(
cop
(
cv
x7
)
(
cv
x8
)
)
ccgr
)
(
wbr
(
cop
(
cv
x4
)
(
cv
x5
)
)
(
cop
(
cv
x8
)
(
cv
x9
)
)
ccgr
)
)
(
wa
(
wbr
(
cop
(
cv
x3
)
(
cv
x6
)
)
(
cop
(
cv
x7
)
(
cv
x10
)
)
ccgr
)
(
wbr
(
cop
(
cv
x4
)
(
cv
x6
)
)
(
cop
(
cv
x8
)
(
cv
x10
)
)
ccgr
)
)
)
)
(
λ x10 .
cfv
(
cv
x2
)
cee
)
)
(
λ x9 .
cfv
(
cv
x2
)
cee
)
)
(
λ x8 .
cfv
(
cv
x2
)
cee
)
)
(
λ x7 .
cfv
(
cv
x2
)
cee
)
)
(
λ x6 .
cfv
(
cv
x2
)
cee
)
)
(
λ x5 .
cfv
(
cv
x2
)
cee
)
)
(
λ x4 .
cfv
(
cv
x2
)
cee
)
)
(
λ x3 .
cfv
(
cv
x2
)
cee
)
)
(
λ x2 .
cn
)
)
)
.
Assume H0:
wceq
ccart
(
cdif
(
cxp
(
cxp
cvv
cvv
)
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
(
cpprod
cep
cep
)
cvv
)
)
)
)
.
Assume H1:
wceq
cimg
(
ccom
(
cimage
(
cres
(
ccom
c2nd
c1st
)
(
cres
c1st
(
cxp
cvv
cvv
)
)
)
)
ccart
)
.
Assume H2:
wceq
cdomain
(
cimage
(
cres
c1st
(
cxp
cvv
cvv
)
)
)
.
Assume H3:
wceq
crange
(
cimage
(
cres
c2nd
(
cxp
cvv
cvv
)
)
)
.
Assume H4:
wceq
ccup
(
cdif
(
cxp
(
cxp
cvv
cvv
)
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
(
cun
(
ccom
(
ccnv
c1st
)
cep
)
(
ccom
(
ccnv
c2nd
)
cep
)
)
cvv
)
)
)
)
.
Assume H5:
wceq
ccap
(
cdif
(
cxp
(
cxp
cvv
cvv
)
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
...
)
...
)
)
)
.
...
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