reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);
reserve a,b for Real;

theorem Th13:
  for A being set, f being Element of Funcs(A,REAL) holds
    (RealFuncExtMult A).(a,(RealFuncExtMult A).(b,f)) =
      (RealFuncExtMult A).(a*b,f)
proof
  let A be set, f be Element of Funcs(A,REAL);
  reconsider aa=a, bb=b as Element of REAL by XREAL_0:def 1;
  per cases;
  suppose
A1: A = {};
    (RealFuncExtMult A).(b,f) = multreal[;](b,f) by Def3;
    hence (RealFuncExtMult A).(a,(RealFuncExtMult A).(b,f))
       = multreal[;](aa,multreal[;](bb,f)) by Def3
      .= multreal[;](a*b,f) by A1
      .= (RealFuncExtMult A).(a*b,f) by Def3;
  end;
  suppose
    A <> {};
    then reconsider A as non empty set;
    reconsider f as Element of Funcs(A,REAL);
    now
      let x be Element of A;
      thus ((RealFuncExtMult A).[aa,(RealFuncExtMult A).[bb,f]]).x
         = aa*(((RealFuncExtMult A).[bb,f]).x) by Th4
        .= aa*(bb*(f.x)) by Th4
        .= (aa*bb)*(f.x)
        .= ((RealFuncExtMult A).[aa*bb,f]).x by Th4;
    end;
    hence thesis by FUNCT_2:63;
  end;
end;
