reserve a,b,c,d,e,z for object, A,B,C,D,E for set;

theorem
  z in [:A,B,C:] implies z`1_3 in A & z`2_3 in B & z`3_3 in C
proof
  assume
A1: z in [:A,B,C:];
  then
A2: C is non empty by MCART_1:31;
A3: A is non empty & B is non empty by A1,MCART_1:31;
  then consider
  a being Element of A, b being Element of B, c being Element of C
  such that
A4: z = [a,b,c] by A1,A2,DOMAIN_1:3;
A5: z`3_3 = c by A4;
  z`1_3 = a & z`2_3 = b by A4;
  hence thesis by A3,A2,A5;
end;
