let D1, D2 be set ; :: thesis: ( ( for o being object holds
( o in D1 iff ex x1, y1 being Surreal st
( 0_No < x1 & x1 in X & y1 in Y & o = (1_No + ((x1 - x) * y1)) * (x1 ") ) ) ) & ( for o being object holds
( o in D2 iff ex x1, y1 being Surreal st
( 0_No < x1 & x1 in X & y1 in Y & o = (1_No + ((x1 - x) * y1)) * (x1 ") ) ) ) implies D1 = D2 )
assume that
A10: for o being object holds
( o in D1 iff ex x1, y1 being Surreal st
( 0_No < x1 & x1 in X & y1 in Y & o = (1_No + ((x1 - x) * y1)) * (x1 ") ) ) and
A11: for o being object holds
( o in D2 iff ex x1, y1 being Surreal st
( 0_No < x1 & x1 in X & y1 in Y & o = (1_No + ((x1 - x) * y1)) * (x1 ") ) ) ; :: thesis: D1 = D2

now :: thesis: for o being object holds
( o in D1 iff o in D2 )

let o be object ; :: thesis: ( o in D1 iff o in D2 )
( o in D1 iff ex x1, y1 being Surreal st
( 0_No < x1 & x1 in X & y1 in Y & o = (1_No + ((x1 - x) * y1)) * (x1 ") ) ) by A10;
hence ( o in D1 iff o in D2 ) by A11; :: thesis: verum

end;

hence D1 = D2 by TARSKI:2; :: thesis: verum