let X1, X2 be Subset of (R_VectorSpace_of_BoundedFunctions (X,Y)); :: thesis: ( ( for x being set holds
( x in X1 iff ex f being continuous PartFunc of REAL,Y st
( x = f & dom f = X ) ) ) & ( for x being set holds
( x in X2 iff ex f being continuous PartFunc of REAL,Y st
( x = f & dom f = X ) ) ) implies X1 = X2 )
assume that
A4: for x being set holds
( x in X1 iff ex f being continuous PartFunc of REAL,Y st
( x = f & dom f = X ) ) and
A5: for x being set holds
( x in X2 iff ex f being continuous PartFunc of REAL,Y st
( x = f & dom f = X ) ) ; :: thesis: X1 = X2
for x being object st x in X2 holds
x in X1

proof

let x be object ; :: thesis: ( x in X2 implies x in X1 )
assume x in X2 ; :: thesis: x in X1
then ex f being continuous PartFunc of REAL,Y st
( x = f & dom f = X ) by A5;
hence x in X1 by A4; :: thesis: verum

end;

then A6: X2 c= X1 by TARSKI:def 3;
for x being object st x in X1 holds
x in X2

proof

let x be object ; :: thesis: ( x in X1 implies x in X2 )
assume x in X1 ; :: thesis: x in X2
then ex f being continuous PartFunc of REAL,Y st
( x = f & dom f = X ) by A4;
hence x in X2 by A5; :: thesis: verum

end;

then X1 c= X2 by TARSKI:def 3;
hence X1 = X2 by A6, XBOOLE_0:def 10; :: thesis: verum