deffunc H₁( set ) -> set = (t1 . $1) mod (t2 . $1);
consider f being Function such that
A7: ( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in (dom t1) /\ (dom t2) holds
f . x = H₁(x) ) ) from FUNCT_1:sch 3();
f is integer-yielding

proof

let x be set ; :: according to VALUED_0:def 11 :: thesis: ( not x in proj1 f or f . x is integer )
assume x in dom f ; :: thesis: f . x is integer
then f . x = H₁(x) by A7;
hence f . x is integer ; :: thesis: verum

end;

hence ex f being integer-yielding Function st
( dom f = (dom t1) /\ (dom t2) & ( for x being set st x in dom f holds
f . x = H₁(x) ) ) by A7; :: thesis: for b₁, b₂ being integer-yielding Function st dom b₁ = (dom t1) /\ (dom t2) & ( for s being set st s in dom b₁ holds
b₁ . s = (t1 . s) mod (t2 . s) ) & dom b₂ = (dom t1) /\ (dom t2) & ( for s being set st s in dom b₂ holds
b₂ . s = (t1 . s) mod (t2 . s) ) holds
b₁ = b₂
let f1, f2 be integer-yielding Function; :: thesis: ( dom f1 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f1 holds
f1 . s = (t1 . s) mod (t2 . s) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being set st s in dom f2 holds
f2 . s = (t1 . s) mod (t2 . s) ) implies f1 = f2 )
assume that
A8: dom f1 = (dom t1) /\ (dom t2) and
A9: for s being set st s in dom f1 holds
f1 . s = H₁(s) and
A10: dom f2 = (dom t1) /\ (dom t2) and
A11: for s being set st s in dom f2 holds
f2 . s = H₁(s) ; :: thesis: f1 = f2

now

let s be set ; :: thesis: ( s in (dom t1) /\ (dom t2) implies f1 . s = f2 . s )
assume A12: s in (dom t1) /\ (dom t2) ; :: thesis: f1 . s = f2 . s
hence f1 . s = H₁(s) by A8, A9
.= f2 . s by A10, A11, A12 ;
:: thesis: verum

end;

hence f1 = f2 by A8, A10, FUNCT_1:2; :: thesis: verum