theorem Th40: f is X*-valued implies f.x in X*
proof
assume f is X*-valued; then
A1: rng f c= X*;
per cases;
suppose A2: x in dom f; then reconsider D=dom f as non empty set;
reconsider e=x as Element of D by A2;
reconsider ff=f as Function of D, X* by FUNCT_2:2, A1;
ff.e is Element of X*; hence thesis;
end;
suppose not x in dom f; then f.x = {} by FUNCT_1:def 2;
hence thesis by FINSEQ_1:49;
end;
end;
