defpred S₁[ object ] means $1 is Lipschitzian LinearOperator of X,Y;
consider IT being set such that
A1: for x being object holds
( x in IT iff ( x in LinearOperators (X,Y) & S₁[x] ) ) from XBOOLE_0:sch 1();
take IT ; :: thesis: ( IT is Subset of (R_VectorSpace_of_LinearOperators (X,Y)) & ( for x being set holds
( x in IT iff x is Lipschitzian LinearOperator of X,Y ) ) )
for x being object st x in IT holds
x in LinearOperators (X,Y) by A1;
hence IT is Subset of (R_VectorSpace_of_LinearOperators (X,Y)) by TARSKI:def 3; :: thesis: for x being set holds
( x in IT iff x is Lipschitzian LinearOperator of X,Y )
let x be set ; :: thesis: ( x in IT iff x is Lipschitzian LinearOperator of X,Y )
thus ( x in IT implies x is Lipschitzian LinearOperator of X,Y ) by A1; :: thesis: ( x is Lipschitzian LinearOperator of X,Y implies x in IT )
assume A2: x is Lipschitzian LinearOperator of X,Y ; :: thesis: x in IT
then x in LinearOperators (X,Y) by Def6;
hence x in IT by A1, A2; :: thesis: verum