reserve x,y for Element of REAL;
reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem
  for x,y holds inv *(x,y) = *(inv x, inv y)
proof
  reconsider jj = 1 as Element of REAL by NUMBERS:19;
  let x,y;
  per cases;
  suppose
A1: x = 0 or y = 0;
    then
A2: inv x = 0 or inv y = 0 by Def4;
    *(x,y) = 0 by A1,Th12;
    hence inv *(x,y) = 0 by Def4
      .= *(inv x, inv y) by A2,Th12;
  end;
  suppose that
A3: x <> 0 and
A4: y <> 0;
A5: *(x,y) <> 0 by A3,A4,Th21;
A6: *(x,inv x) = 1 by A3,Def4;
A7: *(y,inv y) = 1 by A4,Def4;
    *(*(x,y),*(inv x, inv y)) = *(*(*(x,y),inv x), inv y) by Th13
      .= *(*(jj,y), inv y) by A6,Th13
      .= 1 by A7,Th19;
    hence thesis by A5,Def4;
  end;
end;
