defpred S_{1}[ 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 & S_{1}[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

consider IT being set such that

A1: for x being object holds

( x in IT iff ( x in LinearFunctionals X & S

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