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
  min(f,g\h) c= min(f,g)\min(f,h)
proof
  let c;
  (1_minus h).c <= max((1_minus f).c,(1_minus h).c) by XXREAL_0:25;
  then min(min(f,g).c,(1_minus h).c) <= min(min(f,g).c,max((1_minus f).c,(
  1_minus h).c)) by XXREAL_0:18;
  then
  min(min(f,g).c,(1_minus h).c) <= min(min(f,g).c,max(1_minus f,1_minus h)
  .c) by FUZZY_1:5;
  then
A1: min(min(f,g).c,(1_minus h).c) <= min(min(f,g),max(1_minus f,1_minus h)).
  c by FUZZY_1:5;
  min(f,g\h).c = min(min(f,g),1_minus h).c by FUZZY_1:7
    .= min(min(f,g).c,(1_minus h).c) by FUZZY_1:5;
  hence thesis by A1,FUZZY_1:11;
end;
