reserve C for Category,
  C1,C2 for Subcategory of C;

theorem Th5:
  the carrier of C1 meets the carrier of C2 implies C1 /\ C2 = C2 /\ C1
proof
  assume (the carrier of C1) /\ the carrier of C2 <> {};
  then reconsider O = (the carrier of C1) /\ the carrier of C2 as non empty
  set;
  set o = the Element of O;
  set C12 = C1 /\ C2, C21 = C2 /\ C1;
  set M1 = the carrier' of C1, M2 = the carrier' of C2;
  set O1 = the carrier of C1, O2 = the carrier of C2;
A1: o is Object of C1 by XBOOLE_0:def 4;
A2: o is Object of C2 by XBOOLE_0:def 4;
  then
A3: the carrier of C12 = O by A1,Def2;
A4: the carrier of C21 = O by A1,A2,Def2;
A5: the carrier' of C12 = (the carrier' of C1) /\ the carrier' of C2 by A1,A2
,Def2;
A6: the Source of C21 = (the Source of C2)|M1 by A1,A2,Def2;
A7: the Source of C12 = (the Source of C1)|M2 by A1,A2,Def2;
A8: the Target of C21 = (the Target of C2)|M1 by A1,A2,Def2;
A9: the Target of C12 = (the Target of C1)|M2 by A1,A2,Def2;
A10: the Comp of C21 = (the Comp of C2)||M1 by A1,A2,Def2;
A11: the Comp of C12 = (the Comp of C1)||M2 by A1,A2,Def2;
A12: the Source of C1 = (the Source of C)|M1 by NATTRA_1:8;
A13: the Target of C1 = (the Target of C)|M1 by NATTRA_1:8;
A14: the Source of C2 = (the Source of C)|M2 by NATTRA_1:8;
A15: the Target of C2 = (the Target of C)|M2 by NATTRA_1:8;
A16: the Source of C12 = (the Source of C)|(M1 /\ M2) by A7,A12,RELAT_1:71
    .= the Source of C21 by A6,A14,RELAT_1:71;
A17: the Target of C12 = (the Target of C)|(M1 /\ M2) by A9,A13,RELAT_1:71
    .= the Target of C21 by A8,A15,RELAT_1:71;
 the Comp of C12 = (the Comp of C)||M1||M2 by A11,NATTRA_1:8
    .= (the Comp of C)|([:M1,M1:] /\ [:M2,M2:]) by RELAT_1:71
    .= (the Comp of C)||M2||M1 by RELAT_1:71
    .= the Comp of C21 by A10,NATTRA_1:8;
  hence thesis by A1,A2,A3,A4,A5,A16,A17,Def2;
end;
