reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem
  f \+\ g c= max(f,g)\min(f,g)
proof
  let c;
  (max(f,g)\min(f,g)).c = min(max(f,g),max(1_minus(f),1_minus(g))).c by
FUZZY_1:11
    .= max(min(max(f,g),1_minus f),min(max(f,g),1_minus g)).c by FUZZY_1:9
    .= max(max(min(1_minus f,f),min(1_minus f,g)),min(1_minus g,max(f,g))).c
  by FUZZY_1:9
    .= max(max(min(1_minus f,f),min(1_minus f,g)) ,max(min(1_minus g,f),min(
  1_minus g,g))).c by FUZZY_1:9
    .= max(max(max(min(1_minus f,f),min(1_minus f,g)),min(1_minus g,f)), min
  (1_minus g,g)).c by FUZZY_1:7
    .= max(max(min(1_minus f,f),max(min(1_minus f,g),min(1_minus g,f))), min
  (1_minus g,g)).c by FUZZY_1:7
    .= max(max(min(1_minus f,g),min(1_minus g,f)), max(min(1_minus f,f),min(
  1_minus g,g))).c by FUZZY_1:7
    .= max((f \+\ g).c, max(min(1_minus f,f),min(1_minus g,g)).c) by FUZZY_1:5;
  hence thesis by XXREAL_0:25;
end;
