let X1, X2 be Subset of (RAlgebra NAT); :: thesis: ( ( for x being object holds
( x in X1 iff x is polynomially-abs-bounded Function of NAT,REAL ) ) & ( for x being object holds
( x in X2 iff x is polynomially-abs-bounded Function of NAT,REAL ) ) implies X1 = X2 )
assume that
A2: for x being object holds
( x in X1 iff x is polynomially-abs-bounded Function of NAT,REAL ) and
A3: for x being object holds
( x in X2 iff x is polynomially-abs-bounded Function of NAT,REAL ) ; :: thesis: X1 = X2
thus X1 c= X2 :: according to XBOOLE_0:def 10 :: thesis: X2 c= X1

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X1 or x in X2 )
assume x in X1 ; :: thesis: x in X2
then x is polynomially-abs-bounded Function of NAT,REAL by A2;
hence x in X2 by A3; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X2 or x in X1 )
assume x in X2 ; :: thesis: x in X1
then x is polynomially-abs-bounded Function of NAT,REAL by A3;
hence x in X1 by A2; :: thesis: verum