reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem Th50:
  a>0 & a<>1 & c <>d implies a to_power c <> a to_power d
proof
  assume that
A1: a>0 and
A2: a<>1 and
A3: c <>d;
 now per cases by A3,XXREAL_0:1;
    suppose
A4:   c <d;
   now per cases by A2,XXREAL_0:1;
        suppose
       a<1;
          hence thesis by A1,A4,Th40;
        end;
        suppose
       a>1;
          hence thesis by A4,Th39;
        end;
      end;
      hence thesis;
    end;
    suppose
A5:   c>d;
  now per cases by A2,XXREAL_0:1;
        suppose
      a<1;
          hence thesis by A1,A5,Th40;
        end;
        suppose
      a>1;
          hence thesis by A5,Th39;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
