set A = { f where f is continuous RealMap of X : verum } ;
{ f where f is continuous RealMap of X : verum } c= Funcs ( the carrier of X,REAL)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { f where f is continuous RealMap of X : verum } or x in Funcs ( the carrier of X,REAL) )
assume x in { f where f is continuous RealMap of X : verum } ; :: thesis: x in Funcs ( the carrier of X,REAL)
then ex f being continuous RealMap of X st x = f ;
hence x in Funcs ( the carrier of X,REAL) by FUNCT_2:8; :: thesis: verum

end;

hence { f where f is continuous RealMap of X : verum } is Subset of (RAlgebra the carrier of X) ; :: thesis: verum