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 implies max(f,g\f) c= g
proof
  assume
A1: f.c <= g.c;
  let c;
A2: f.c <= g.c by A1;
  max(f,g\f).c = min(max(f,g),max(f,1_minus f)).c by FUZZY_1:9
    .= min(max(f,g).c,max(f,1_minus f).c) by FUZZY_1:5
    .= min(max(f.c,g.c),max(f,1_minus f).c) by FUZZY_1:5
    .= min(g.c,max(f,1_minus f).c) by A2,XXREAL_0:def 10;
  hence thesis by XXREAL_0:17;
end;
