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

theorem Th31:
  for C being Category, a being Object of C, m being Morphism of a-SliceCat C
  holds m = [[m`11,m`12],m`2] & dom m`2 = cod m`11 & m`2(*)m`11 = m`12 &
  dom m = m`11 & cod m = m`12
proof
  let C be Category, o be Object of C, m be Morphism of o-SliceCat C;
  consider a,b being Element of o Hom, f being Morphism of C such that
A1: m = [[a,b],f] and
A2: dom f = cod a and
A3: f(*)a = b by Def12;
A4: m`11 = a by A1,MCART_1:85;
  m`12 = b by A1,MCART_1:85;
  hence thesis by A1,A2,A3,A4,Th2;
end;
