reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;

theorem Th2:
  h = (FuncExtMult(X,Y)).[a,f] iff for x being Element of X holds h
  .x = a*(f.x)
proof
  thus h = (FuncExtMult(X,Y)).[a,f] implies for x being Element of X holds h.x
  = a*(f.x) by Def2;
 reconsider a as Element of REAL by XREAL_0:def 1;
  now
    assume
A1: for x being Element of X holds h.x = a*(f.x);
    for x being Element of X holds h.x = ((FuncExtMult(X,Y)).[a,f]).x
    proof
      let x be Element of X;
      thus h.x = a*(f.x) by A1
        .= ((FuncExtMult(X,Y)).[a,f]).x by Def2;
    end;
    hence h = (FuncExtMult(X,Y)).[a,f] by FUNCT_2:63;
  end;
  hence thesis;
end;
