defpred S₁[ set ] means $1 is bounded LinearOperator of X,Y;
consider IT being set such that
A1: for x being set 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 bounded LinearOperator of X,Y ) ) )
for x being set 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 bounded LinearOperator of X,Y )
let x be set ; :: thesis: ( x in IT iff x is bounded LinearOperator of X,Y )
thus ( x in IT implies x is bounded LinearOperator of X,Y ) by A1; :: thesis: ( x is bounded LinearOperator of X,Y implies x in IT )
assume A2: x is bounded LinearOperator of X,Y ; :: thesis: x in IT
then x in LinearOperators X,Y by Def7;
hence x in IT by A1, A2; :: thesis: verum