defpred S₁[ object ] means $1 is Lipschitzian BilinearOperator of X,Y,Z;
consider IT being set such that
A1: for x being object holds
( x in IT iff ( x in BilinearOperators (X,Y,Z) & S₁[x] ) ) from XBOOLE_0:sch 1();
take IT ; :: thesis: ( IT is Subset of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) & ( for x being set holds
( x in IT iff x is Lipschitzian BilinearOperator of X,Y,Z ) ) )
for x being object st x in IT holds
x in BilinearOperators (X,Y,Z) by A1;
hence IT is Subset of (R_VectorSpace_of_BilinearOperators (X,Y,Z)) by TARSKI:def 3; :: thesis: for x being set holds
( x in IT iff x is Lipschitzian BilinearOperator of X,Y,Z )
let x be set ; :: thesis: ( x in IT iff x is Lipschitzian BilinearOperator of X,Y,Z )
thus ( x in IT implies x is Lipschitzian BilinearOperator of X,Y,Z ) by A1; :: thesis: ( x is Lipschitzian BilinearOperator of X,Y,Z implies x in IT )
assume x is Lipschitzian BilinearOperator of X,Y,Z ; :: thesis: x in IT
hence x in IT by A1, EQSET; :: thesis: verum