reserve c,c1,c2,x,y,z,z1,z2 for set;
reserve C1,C2,C3 for non empty set;

theorem Th3:
  for f be RMembership_Func of C1,C2, c be set holds 0 <= f.c & f.c <= 1
proof
  let f being RMembership_Func of C1,C2;
  let c be set;
  per cases;
  suppose
    c in [:C1,C2:];
    then reconsider c as Element of [:C1,C2:];
A1: f.c <= (Umf(C1,C2)).c by FUZZY_2:52;
    (Zmf(C1,C2)).c <= f.c by FUZZY_2:52;
    hence thesis by A1,FUNCT_3:def 3;
  end;
  suppose
A2: not c in [:C1,C2:];
    dom f = [:C1,C2:] by FUNCT_2:def 1;
    hence thesis by A2,FUNCT_1:def 2;
  end;
end;
