let I be set ; for A being ManySortedSet of I
for B, C being V2() ManySortedSet of I
for F being ManySortedFunction of A,B
for G being ManySortedFunction of B,C
for X being ManySortedSubset of A holds (G ** F) || X = G ** (F || X)
let A be ManySortedSet of I; for B, C being V2() ManySortedSet of I
for F being ManySortedFunction of A,B
for G being ManySortedFunction of B,C
for X being ManySortedSubset of A holds (G ** F) || X = G ** (F || X)
let B, C be V2() ManySortedSet of I; for F being ManySortedFunction of A,B
for G being ManySortedFunction of B,C
for X being ManySortedSubset of A holds (G ** F) || X = G ** (F || X)
let F be ManySortedFunction of A,B; for G being ManySortedFunction of B,C
for X being ManySortedSubset of A holds (G ** F) || X = G ** (F || X)
let G be ManySortedFunction of B,C; for X being ManySortedSubset of A holds (G ** F) || X = G ** (F || X)
let X be ManySortedSubset of A; (G ** F) || X = G ** (F || X)
now for i being object st i in I holds
((G ** F) || X) . i = (G ** (F || X)) . ilet i be
object ;
( i in I implies ((G ** F) || X) . i = (G ** (F || X)) . i )assume A1:
i in I
;
((G ** F) || X) . i = (G ** (F || X)) . ithen reconsider gf =
(G ** F) . i as
Function of
(A . i),
(C . i) by PBOOLE:def 15;
reconsider fx =
(F || X) . i as
Function of
(X . i),
(B . i) by A1, PBOOLE:def 15;
reconsider g =
G . i as
Function of
(B . i),
(C . i) by A1, PBOOLE:def 15;
reconsider f =
F . i as
Function of
(A . i),
(B . i) by A1, PBOOLE:def 15;
thus ((G ** F) || X) . i =
gf | (X . i)
by A1, MSAFREE:def 1
.=
(g * f) | (X . i)
by A1, MSUALG_3:2
.=
g * (f | (X . i))
by RELAT_1:83
.=
g * fx
by A1, MSAFREE:def 1
.=
(G ** (F || X)) . i
by A1, MSUALG_3:2
;
verum end;
hence
(G ** F) || X = G ** (F || X)
; verum