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
  a>0 & a<1 & b>0 & c>b implies log(a,c) < log(a,b)
proof
  assume that
A1: a>0 & a<1 and
A2: b>0 and
A3: c>b and
A4: log(a,c) >= log(a,b);
A5: a to_power log(a,b) = b by A1,A2,Def3;
A6: a to_power log(a,c) = c by A1,A2,A3,Def3;
 now per cases by A4,XXREAL_0:1;
    suppose
   log(a,c)>log(a,b);
      hence contradiction by A1,A3,A5,A6,Th40;
    end;
    suppose
   log(a,c) = log(a,b);
      hence contradiction by A1,A2,A3,A6,Def3;
    end;
  end;
  hence contradiction;
end;
