
theorem Th3:
  for C1 being Category, C2 being Subcategory of C1 st C1 is Subcategory of C2
  holds the CatStr of C1 = the CatStr of C2
proof
  let C1 be Category, C2 be Subcategory of C1;
  assume
A1: C1 is Subcategory of C2;
  then
A2: the carrier of C1 c= the carrier of C2 by CAT_2:def 4;
 the carrier of C2 c= the carrier of C1 by CAT_2:def 4;
  then
A3: the carrier of C1 = the carrier of C2 by A2;
A4: the carrier' of C1 c= the carrier' of C2 by A1,CAT_2:7;
 the carrier' of C2 c= the carrier' of C1 by CAT_2:7;
  then
A5: the carrier' of C1 = the carrier' of C2 by A4;
A6: the Comp of C1 c= the Comp of C2 by A1,CAT_2:def 4;
  the Comp of C2 c= the Comp of C1 by CAT_2:def 4;
  then
A7: the Comp of C1 = the Comp of C2 by A6;
  now
    let m1 be Morphism of C1;
    reconsider m2 = m1 as Morphism of C2 by A4;
    thus (the Source of C1).m1 = dom m1 .= dom m2 by CAT_2:9
      .= (the Source of C2).m1;
  end;
  then
A8: the Source of C1 = the Source of C2 by A3,A5,FUNCT_2:63;
  now
    let m1 be Morphism of C1;
    reconsider m2 = m1 as Morphism of C2 by A4;
    thus (the Target of C1).m1 = cod m1 .= cod m2 by CAT_2:9
      .= (the Target of C2).m1;
  end;
  then
 the Target of C1 = the Target of C2 by A3,A5,FUNCT_2:63;
  hence thesis by A3,A5,A7,A8;
end;
