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 c= g & h c= h1 implies f\h1 c= g\h
proof
  assume that
A1: f.c <= g.c and
A2: h.c <= h1.c;
  let c;
  h.c <= h1.c by A2;
  then
A3: 1-h.c >= 1-h1.c by XREAL_1:10;
  f.c <= g.c by A1;
  then min(f.c,1-h1.c) <= min(g.c,1- h.c) by A3,XXREAL_0:18;
  then min(f.c,(1_minus h1).c) <= min(g.c,1- h.c) by FUZZY_1:def 5;
  then min(f.c,(1_minus h1).c) <= min(g.c,(1_minus h).c) by FUZZY_1:def 5;
  then min(f,1_minus h1).c <= min(g.c,(1_minus h).c) by FUZZY_1:5;
  hence thesis by FUZZY_1:5;
end;
