defpred S_{1}[ object ] means $1 is linear-Functional of X;

consider IT being set such that

A1: for x being object holds

( x in IT iff ( x in Funcs ( the carrier of X,REAL) & S_{1}[x] ) )
from XBOOLE_0:sch 1();

for x being object st x in IT holds

x in Funcs ( the carrier of X,REAL) by A1;

then reconsider IT = IT as Subset of (RealVectSpace the carrier of X) by TARSKI:def 3;

take IT ; :: thesis: for x being object holds

( x in IT iff x is linear-Functional of X )

let x be object ; :: thesis: ( x in IT iff x is linear-Functional of X )

thus ( x in IT implies x is linear-Functional of X ) by A1; :: thesis: ( x is linear-Functional of X implies x in IT )

assume A3: x is linear-Functional of X ; :: thesis: x in IT

then x in Funcs ( the carrier of X,REAL) by FUNCT_2:8;

hence x in IT by A1, A3; :: thesis: verum

consider IT being set such that

A1: for x being object holds

( x in IT iff ( x in Funcs ( the carrier of X,REAL) & S

for x being object st x in IT holds

x in Funcs ( the carrier of X,REAL) by A1;

then reconsider IT = IT as Subset of (RealVectSpace the carrier of X) by TARSKI:def 3;

take IT ; :: thesis: for x being object holds

( x in IT iff x is linear-Functional of X )

let x be object ; :: thesis: ( x in IT iff x is linear-Functional of X )

thus ( x in IT implies x is linear-Functional of X ) by A1; :: thesis: ( x is linear-Functional of X implies x in IT )

assume A3: x is linear-Functional of X ; :: thesis: x in IT

then x in Funcs ( the carrier of X,REAL) by FUNCT_2:8;

hence x in IT by A1, A3; :: thesis: verum