 reserve x,y for Element of [.0,1.];

theorem
  (x = 0 implies I_GG.(x,y) = 1) &
  (x > 0 implies I_GG.(x,y) = min (1, y / x))
  proof
    thus x = 0 implies I_GG.(x,y) = 1
    proof
      assume x = 0; then
      x <= y by XXREAL_1:1;
      hence thesis by FUZIMPL1:def 19;
    end;
    assume
S0: x > 0;
A1: y >= 0 by XXREAL_1:1;
    per cases;
    suppose
A2:   x <= y; then
S1:   I_GG.(x,y) = 1 by FUZIMPL1:def 19;
      y / x >= 1 by A2,S0,XREAL_1:181;
      hence thesis by S1,XXREAL_0:def 9;
    end;
    suppose
S2:   x > y; then
      I_GG.(x,y) = y / x by FUZIMPL1:def 19;
      hence thesis by S2,A1,XREAL_1:183,XXREAL_0:def 9;
    end;
  end;
