reserve C for category,
  o1, o2, o3 for Object of C;

theorem
  o1 is initial & o1, o2 are_iso implies o2 is initial
proof
  assume that
A1: o1 is initial and
A2: o1, o2 are_iso;
  for o3 being Object of C holds ex M being Morphism of o2, o3 st M in <^
  o2,o3^> & for v being Morphism of o2, o3 st v in <^o2,o3^> holds M = v
  proof
    consider f being Morphism of o1, o2 such that
A3: f is iso by A2;
A4: f * f" = idm o2 by A3;
    let o3 be Object of C;
    consider u being Morphism of o1, o3 such that
A5: u in <^o1,o3^> and
A6: for M1 being Morphism of o1, o3 st M1 in <^o1,o3^> holds u = M1 by A1,
ALTCAT_3:25;
    take u * f";
A7: <^o2,o1^> <> {} by A2;
    then
A8: <^o2,o3^> <> {} by A5,ALTCAT_1:def 2;
    hence u * f" in <^o2,o3^>;
A9: <^o1,o2^> <> {} by A2;
    let v be Morphism of o2, o3 such that
    v in <^o2,o3^>;
    v * f = u by A5,A6;
    hence thesis by A4,A9,A7,A8,Th2;
  end;
  hence thesis by ALTCAT_3:25;
end;
