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

theorem Th22:
  for f be RMembership_Func of C2,C3 holds Zmf(C1,C2)(#)f = Zmf(C1 ,C3)
proof
  let f be RMembership_Func of C2,C3;
A1: dom(Zmf(C1,C3)) = [:C1,C3:] by FUNCT_2:def 1;
A2: for c being Element of [:C1,C3:] st c in [:C1,C3:] holds (Zmf(C1,C2)(#)f
  ).c = Zmf(C1,C3).c
  proof
    let c be Element of [:C1,C3:];
    consider x,z being object such that
A3: x in C1 and
A4: z in C3 and
A5: c = [x,z] by ZFMISC_1:def 2;
    reconsider z,x as set by TARSKI:1;
    (Zmf(C1,C2)(#)f).c = (Zmf(C1,C2)(#)f).(x,z) by A5
      .= upper_bound rng min(Zmf(C1,C2),f,x,z) by A5,Def3
      .= Zmf(C1,C3).c by A3,A4,A5,Lm7;
    hence thesis;
  end;
  dom(Zmf(C1,C2)(#)f) = [:C1,C3:] by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
