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= min(f,g)
proof
  let c;
A1: min(f,g).c = min(f.c,g.c) by FUZZY_1:5;
  per cases by A1,XXREAL_0:15;
  suppose
A2: min(f,g).c = f.c;
A3: f.c >= 0 by Th1;
    g.c <= 1 by Th1;
    then (g.c)*(f.c) <= 1*(f.c) by A3,XREAL_1:64;
    hence thesis by A2,Def2;
  end;
  suppose
A4: min(f,g).c = g.c;
A5: g.c >= 0 by Th1;
    f.c <= 1 by Th1;
    then (f.c)*(g.c) <= 1*(g.c) by A5,XREAL_1:64;
    hence thesis by A4,Def2;
  end;
end;
