defpred S₁[ object ] means $1 is Lipschitzian linear-Functional of X;
consider IT being set such that
A1: for x being object holds
( x in IT iff ( x in LinearFunctionals X & S₁[x] ) ) from XBOOLE_0:sch 1();
take IT ; :: thesis: ( IT is Subset of (X *') & ( for x being set holds
( x in IT iff x is Lipschitzian linear-Functional of X ) ) )
for x being object st x in IT holds
x in LinearFunctionals X by A1;
hence IT is Subset of (X *') by TARSKI:def 3; :: thesis: for x being set holds
( x in IT iff x is Lipschitzian linear-Functional of X )
let x be set ; :: thesis: ( x in IT iff x is Lipschitzian linear-Functional of X )
thus ( x in IT implies x is Lipschitzian linear-Functional of X ) by A1; :: thesis: ( x is Lipschitzian linear-Functional of X implies x in IT )
assume A2: x is Lipschitzian linear-Functional of X ; :: thesis: x in IT
then x in LinearFunctionals X by Def7;
hence x in IT by A1, A2; :: thesis: verum