defpred S₁[ 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) & S₁[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