consider X being set such that
A1: for x being object holds
( x in X iff ( x in Funcs ( the carrier of I[01], the carrier of T) & S₁[x] ) ) from XBOOLE_0:sch 1();
take X ; :: thesis: for x being object holds
( x in X iff x is Path of t1,t2 )
let x be object ; :: thesis: ( x in X iff x is Path of t1,t2 )
thus ( x in X implies x is Path of t1,t2 ) by A1; :: thesis: ( x is Path of t1,t2 implies x in X )
assume A2: x is Path of t1,t2 ; :: thesis: x in X
then x in Funcs ( the carrier of I[01], the carrier of T) by FUNCT_2:8;
hence x in X by A1, A2; :: thesis: verum