reserve C for CatStr;
reserve f,g for Morphism of C;
reserve C for non void non empty CatStr,
  f,g for Morphism of C,
  a,b,c,d for Object of C;
reserve o,m for set;
reserve B,C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,f1,f2,g,g1,g2 for Morphism of C;
reserve f,f1,f2 for Morphism of a,b;
reserve f9 for Morphism of b,a;
reserve g for Morphism of b,c;
reserve h,h1,h2 for Morphism of c,d;

theorem
  a is initial & a,b are_isomorphic implies b is initial
proof
  assume that
A1: a is initial;
  given f such that
A2: f is invertible;
A3: Hom(b,a) <> {} by A2;
  let c;
  consider h being Morphism of a,c such that
A4: for g being Morphism of a,c holds h = g by A1;
  Hom(a,c) <> {} by A1;
  hence
A5: Hom(b,c) <> {} by A3,Th19;
  consider f9 such that
A6: f*f9 = id b and
  f9*f = id a by A2;
A7: Hom(a,b) <> {} by A2;
  take h*f9;
  let h9 be Morphism of b,c;
  thus h9 = h9*(f*f9) by A6,A5,Th24
    .= (h9*f)*f9 by A3,A5,A7,Th21
    .= h*f9 by A4;
end;
