reserve a,b,c,d for Real;
reserve r,s for Real;

theorem
  a <= 1 & 0 <= b & b <= 1 & a*b = 1 implies a = 1
proof
  assume that
A1: a<=1 and
A2: 0<=b and
A3: b<=1 and
A4: a*b=1;
  now
    per cases by A2;
    case
      b=0;
      hence contradiction by A4;
    end;
    case
A5:   b>0;
      then a=b" by A4,XCMPLX_0:def 7;
      then a>=1" by A3,A5,Lm32;
      hence thesis by A1,XXREAL_0:1;
    end;
  end;
  hence thesis;
end;
