let F, G be Function of [:(Pr D),(FPrg D),(FPrg D):],(FPrg D); :: thesis: ( ( for p being PartialPredicate of D
for f, g being BinominativeFunction of D holds
( dom (F . (p,f,g)) = { d where d is Element of D : ( ( d in dom p & p . d = TRUE & d in dom f ) or ( d in dom p & p . d = FALSE & d in dom g ) ) } & ( for d being object holds
( ( d in dom p & p . d = TRUE & d in dom f implies (F . (p,f,g)) . d = f . d ) & ( d in dom p & p . d = FALSE & d in dom g implies (F . (p,f,g)) . d = g . d ) ) ) ) ) & ( for p being PartialPredicate of D
for f, g being BinominativeFunction of D holds
( dom (G . (p,f,g)) = { d where d is Element of D : ( ( d in dom p & p . d = TRUE & d in dom f ) or ( d in dom p & p . d = FALSE & d in dom g ) ) } & ( for d being object holds
( ( d in dom p & p . d = TRUE & d in dom f implies (G . (p,f,g)) . d = f . d ) & ( d in dom p & p . d = FALSE & d in dom g implies (G . (p,f,g)) . d = g . d ) ) ) ) ) implies F = G )
assume that
A15: for p being PartialPredicate of D
for f, g being BinominativeFunction of D holds
( dom (F . (p,f,g)) = H₁(p,f,g) & ( for d being object holds
( ( S₁[d,p,f] implies (F . (p,f,g)) . d = f . d ) & ( S₂[d,p,g] implies (F . (p,f,g)) . d = g . d ) ) ) ) and
A16: for p being PartialPredicate of D
for f, g being BinominativeFunction of D holds
( dom (G . (p,f,g)) = H₁(p,f,g) & ( for d being object holds
( ( S₁[d,p,f] implies (G . (p,f,g)) . d = f . d ) & ( S₂[d,p,g] implies (G . (p,f,g)) . d = g . d ) ) ) ) ; :: thesis: F = G

now :: thesis: for x1, x2, x3 being object st x1 in Pr D & x2 in FPrg D & x3 in FPrg D holds
F . [x1,x2,x3] = G . [x1,x2,x3]

let x1, x2, x3 be object ; :: thesis: ( x1 in Pr D & x2 in FPrg D & x3 in FPrg D implies F . [x1,x2,x3] = G . [x1,x2,x3] )
assume that
A17: x1 in Pr D and
A18: x2 in FPrg D and
A19: x3 in FPrg D ; :: thesis: F . [x1,x2,x3] = G . [x1,x2,x3]
reconsider p1 = x1 as PartialPredicate of D by A17, PARTFUN1:46;
reconsider f2 = x2, f3 = x3 as BinominativeFunction of D by A18, A19, PARTFUN1:46;
thus F . [x1,x2,x3] = G . [x1,x2,x3] :: thesis: verum

proof

A20: G . [x1,x2,x3] = G . (x1,x2,x3) ;
F . [x1,x2,x3] = F . (x1,x2,x3) ;
then A21: dom (F . [x1,x2,x3]) = H₁(p1,f2,f3) by A15;
hence dom (F . [x1,x2,x3]) = dom (G . [x1,x2,x3]) by A16, A20; :: according to FUNCT_1:def 11 :: thesis: for b₁ being object holds
( not b₁ in dom (F . [x1,x2,x3]) or (F . [x1,x2,x3]) . b₁ = (G . [x1,x2,x3]) . b₁ )
let a be object ; :: thesis: ( not a in dom (F . [x1,x2,x3]) or (F . [x1,x2,x3]) . a = (G . [x1,x2,x3]) . a )
assume a in dom (F . [x1,x2,x3]) ; :: thesis: (F . [x1,x2,x3]) . a = (G . [x1,x2,x3]) . a
then consider d being Element of D such that
A22: a = d and
A23: ( S₁[d,p1,f2] or S₂[d,p1,f3] ) by A21;

per cases ( S₁[d,p1,f2] or S₂[d,p1,f3] ) by A23;

suppose A24: S₁[d,p1,f2] ; :: thesis: (F . [x1,x2,x3]) . a = (G . [x1,x2,x3]) . a

thus (F . [x1,x2,x3]) . a = (F . (p1,f2,f3)) . a
.= f2 . d by A15, A22, A24
.= (G . (p1,f2,f3)) . a by A16, A22, A24
.= (G . [x1,x2,x3]) . a ; :: thesis: verum

end;

suppose A25: S₂[d,p1,f3] ; :: thesis: (F . [x1,x2,x3]) . a = (G . [x1,x2,x3]) . a

thus (F . [x1,x2,x3]) . a = (F . (p1,f2,f3)) . a
.= f3 . d by A15, A22, A25
.= (G . (p1,f2,f3)) . a by A16, A22, A25
.= (G . [x1,x2,x3]) . a ; :: thesis: verum

end;

end;

end;

end;

hence F = G by MULTOP_1:1; :: thesis: verum