
theorem
  for C being AltGraph, o being Object of C holds o is initial iff for
  o1 being Object of C holds ex M being Morphism of o,o1 st M in <^o,o1^> & for
  M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1
proof
  let C be AltGraph, o be Object of C;
  thus o is initial implies for o1 being Object of C holds ex M being Morphism
  of o,o1 st M in <^o,o1^> & for M1 being Morphism of o,o1 st M1 in <^o,o1^>
  holds M = M1
  proof
    assume
A1: o is initial;
    let o1 be Object of C;
    consider M being Morphism of o,o1 such that
A2: M in <^o,o1^> and
A3: <^o,o1^> is trivial by A1;
    ex i be object st <^o,o1^> = { i } by A2,A3,ZFMISC_1:131;
    then <^o,o1^> = {M} by TARSKI:def 1;
    then for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1 by
TARSKI:def 1;
    hence thesis by A2;
  end;
  assume
A4: for o1 being Object of C holds ex M being Morphism of o,o1 st M in
  <^o,o1^> & for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1;
  let o1 be Object of C;
  consider M being Morphism of o,o1 such that
A5: M in <^o,o1^> and
A6: for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1 by A4;
A7: <^o,o1^> c= {M}
  proof
    let x be object;
    assume
A8: x in <^o,o1^>;
    then reconsider M1 = x as Morphism of o,o1;
    M1 = M by A6,A8;
    hence thesis by TARSKI:def 1;
  end;
  {M} c= <^o,o1^>
  by A5,TARSKI:def 1;
  then <^o,o1^> = {M} by A7,XBOOLE_0:def 10;
  hence thesis;
end;
