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

theorem LemmaAB:
  I is satisfying_(EP) satisfying_(OP) implies
     I is satisfying_(I1) satisfying_(I3) satisfying_(I4) satisfying_(I5)
      satisfying_(LB) satisfying_(RB) satisfying_(NC) satisfying_(NP)
        satisfying_(IP)
  proof
    assume
AA: I is satisfying_(EP) satisfying_(OP);
tt: for x1,x2,y being Element of [.0,1.] st
      x1 <= x2 holds I.(x1,y) >= I.(x2,y)
    proof
      let x1,x2,y be Element of [.0,1.];
      assume
Z1:   x1 <= x2;
      I.(x2,I.(I.(x2,y),y)) = I.(I.(x2,y), I.(x2,y)) by AA
         .= 1 by AA; then
      x2 <= I.(I.(x2,y),y) by AA; then
      1 = I.(x1,I.(I.(x2,y),y)) by AA,Z1,XXREAL_0:2
       .= I.(I.(x2,y),I.(x1,y)) by AA;
      hence thesis by AA;
    end;
    for y being Element of [.0,1.] holds I.(1,y) = y
    proof
      let y be Element of [.0,1.];
S1:   1 in [.0,1.] by XXREAL_1:1;
      reconsider i = 1 as Element of [.0,1.] by XXREAL_1:1;
S2:   I.(i,y) in [.0,1.];
      I.(y,I.(1,y)) = I.(1,I.(y,y)) by S1,AA
                   .= I.(1,1) by AA
                   .= 1 by S1,AA; then
Z1:   y <= I.(1,y) by AA,S2;
      I.(i,I.(I.(i,y),y)) = I.(I.(i,y),I.(i,y)) by AA
                         .= 1 by AA; then
Z2:   1 <= I.(I.(i,y),y) by AA;
      1 >= I.(I.(i,y),y) by XXREAL_1:1; then
      I.(I.(i,y),y) = 1 by Z2,XXREAL_0:1; then
      I.(1,y) <= y by AA;
      hence thesis by Z1,XXREAL_0:1;
    end; then
    I is satisfying_(NP);
    hence thesis by AA,tt;
  end;
