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

theorem Th6:
  the carrier of C1 meets the carrier of C2 implies
  C1 /\ C2 is Subcategory of C1 & C1 /\ C2 is Subcategory of C2
proof
  assume
A1: (the carrier of C1) meets the carrier of C2;
  then
A2: (the carrier of C1) /\ the carrier of C2 <> {};
A3: C1 /\ C2 = C2 /\ C1 by A1,Th5;
  now
    let C1,C2 be Subcategory of C;
    assume
A4: (the carrier of C1) /\ the carrier of C2 <> {};
A5: (the carrier of C1) /\ the carrier of C2 c= the carrier of C1 by
XBOOLE_1:17;
A6: (the carrier' of C1) /\ the carrier' of C2 c= the carrier' of C1 by
XBOOLE_1:17;
    reconsider O = (the carrier of C1) /\ the carrier of C2 as non empty set
    by A4;
    set o = the Element of O;
A7: o is Object of C1 by XBOOLE_0:def 4;
A8: o is Object of C2 by XBOOLE_0:def 4;
    then
A9: the carrier of C1/\C2 = (the carrier of C1) /\ the carrier of C2 by A7,Def2
;
A10: the carrier' of C1/\C2 = (the carrier' of C1) /\ the carrier' of C2 by A7
,A8,Def2;
A11: the Source of C1/\C2 = (the Source of C1)|the carrier' of C2 by A7,A8,Def2
;
A12: the Target of C1/\C2 = (the Target of C1)|the carrier' of C2 by A7,A8,Def2
;
A13: the Comp of C1/\C2 = (the Comp of C1)||the carrier' of C2 by A7,A8,Def2;
A14: the Source of C1 = (the Source of C1)|dom the Source of C1;
    dom the Source of C1 = the carrier' of C1 by FUNCT_2:def 1;
    then
A15: the Source of C1/\C2 = (the Source of C1)|the carrier' of C1/\C2
    by A10,A11,A14,RELAT_1:71;
A16: the Target of C1 = (the Target of C1)|dom the Target of C1;
    dom the Target of C1 = the carrier' of C1 by FUNCT_2:def 1;
    then
A17: the Target of C1/\C2 = (the Target of C1)|the carrier' of C1/\C2
    by A10,A12,A16,RELAT_1:71;
A18: for o being Element of C1 st o in O
       holds id o in (the carrier' of C1) /\ the carrier' of C2
    proof let o1 be Element of C1;
     assume
    o1 in O;
      then reconsider o2 = o1 as Element of C2 by XBOOLE_0:def 4;
A19:    the carrier of C1 c= the carrier of C by CAT_2:def 4;
      reconsider o = o1 as Element of C by A19;
A20:    id o1 = id o by CAT_2:def 4;
      id o2 = id o by CAT_2:def 4;
     hence id o1 in (the carrier' of C1) /\ the carrier' of C2
        by A20,XBOOLE_0:def 4;
    end;

    the Comp of C1 = (the Comp of C1)||the carrier' of C1;
    then the Comp of C1/\C2 = (the Comp of C1)|
    ([:the carrier' of C1,the carrier' of C1:] /\
    [:the carrier' of C2,the carrier' of C2:]) by A13,RELAT_1:71;
    then the Comp of C1/\C2 = (the Comp of C1)||the carrier' of C1/\C2
    by A10,ZFMISC_1:100;
    hence C1/\C2 is Subcategory of C1 by A5,A6,A9,A10,A15,A17,A18,NATTRA_1:9;
  end;
  hence thesis by A2,A3;
end;
