deffunc H₁( object ) -> object = IFGT ((t1 . $1),(t2 . $1),1,0);
consider f being Function such that
A25: ( dom f = (dom t1) /\ (dom t2) & ( for x being object st x in (dom t1) /\ (dom t2) holds
f . x = H₁(x) ) ) from FUNCT_1:sch 3();
f is INT -valued

proof

let y be object ; :: according to TARSKI:def 3,RELAT_1:def 19 :: thesis: ( not y in proj2 f or y in INT )
assume y in rng f ; :: thesis: y in INT
then consider x being object such that
A26: x in dom f and
A27: f . x = y by FUNCT_1:def 3;
f . x = H₁(x) by A25, A26;
hence y in INT by A27, INT_1:def 2; :: thesis: verum

end;

hence ex f being INT -valued Function st
( dom f = (dom t1) /\ (dom t2) & ( for x being object st x in dom f holds
f . x = H₁(x) ) ) by A25; :: thesis: for b₁, b₂ being INT -valued Function st dom b₁ = (dom t1) /\ (dom t2) & ( for s being object st s in dom b₁ holds
b₁ . s = IFGT ((t1 . s),(t2 . s),1,0) ) & dom b₂ = (dom t1) /\ (dom t2) & ( for s being object st s in dom b₂ holds
b₂ . s = IFGT ((t1 . s),(t2 . s),1,0) ) holds
b₁ = b₂
let f1, f2 be INT -valued Function; :: thesis: ( dom f1 = (dom t1) /\ (dom t2) & ( for s being object st s in dom f1 holds
f1 . s = IFGT ((t1 . s),(t2 . s),1,0) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being object st s in dom f2 holds
f2 . s = IFGT ((t1 . s),(t2 . s),1,0) ) implies f1 = f2 )
assume that
A28: dom f1 = (dom t1) /\ (dom t2) and
A29: for s being object st s in dom f1 holds
f1 . s = H₁(s) and
A30: dom f2 = (dom t1) /\ (dom t2) and
A31: for s being object st s in dom f2 holds
f2 . s = H₁(s) ; :: thesis: f1 = f2

now :: thesis: for s being object st s in (dom t1) /\ (dom t2) holds
f1 . s = f2 . s

let s be object ; :: thesis: ( s in (dom t1) /\ (dom t2) implies f1 . s = f2 . s )
assume A32: s in (dom t1) /\ (dom t2) ; :: thesis: f1 . s = f2 . s
hence f1 . s = H₁(s) by A28, A29
.= f2 . s by A30, A31, A32 ;
:: thesis: verum

end;

hence f1 = f2 by A28, A30; :: thesis: verum