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 Th12:
  for A being set, f being Element of Funcs(A,REAL) holds
    (RealFuncExtMult A).(1,f) = f
proof
  let A be set, f be Element of Funcs(A,REAL);
  per cases;
  suppose A = {}; then
A1: f = {};
    thus (RealFuncExtMult A).(1,f) = multreal[;](jj,f) by Def3
      .= f by A1;
  end;
  suppose A <> {};
    then reconsider A as non empty set;
    reconsider f as Element of Funcs(A,REAL);
    reconsider g = (RealFuncExtMult A).(jj,f) as Element of Funcs(A,REAL);
    now
      let x be Element of A;
      thus g.x = jj*(f.x) by Th4
        .= f.x;
    end;
    hence thesis by FUNCT_2:63;
  end;
end;
