deffunc H₁( object ) -> object = (t1 . $1) mod (t2 . $1);
consider f being Function such that
A9: ( 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
A10: x in dom f and
A11: f . x = y by FUNCT_1:def 3;
f . x = H₁(x) by A9, A10;
hence y in INT by A11, 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 A9; :: 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 = (t1 . s) mod (t2 . s) ) & dom b₂ = (dom t1) /\ (dom t2) & ( for s being object st s in dom b₂ holds
b₂ . s = (t1 . s) mod (t2 . s) ) 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 = (t1 . s) mod (t2 . s) ) & dom f2 = (dom t1) /\ (dom t2) & ( for s being object st s in dom f2 holds
f2 . s = (t1 . s) mod (t2 . s) ) implies f1 = f2 )
assume that
A12: dom f1 = (dom t1) /\ (dom t2) and
A13: for s being object st s in dom f1 holds
f1 . s = H₁(s) and
A14: dom f2 = (dom t1) /\ (dom t2) and
A15: 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 A16: s in (dom t1) /\ (dom t2) ; :: thesis: f1 . s = f2 . s
hence f1 . s = H₁(s) by A12, A13
.= f2 . s by A14, A15, A16 ;
:: thesis: verum

end;

hence f1 = f2 by A12, A14; :: thesis: verum