reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem Th59:
  for o,m being object for c being Element of 1Cat(o,m) holds
  c is Object of 1Cat(o,m) & c = o & id c = m
  proof
    let o,m be object;
    let c be Element of 1Cat(o,m);
    reconsider o9=o,m9=m as set by TARSKI:1;
    set C = 1Cat(o,m);
A1: C = CatStr(# { o },{ m },  m :-> o , m :-> o,(m,m):->m #)
      by CAT_1:def 11;
    hence c is Object of 1Cat(o,m) & c = o by TARSKI:def 1;
    o9 is Object of 1Cat(o9,m9) by A1,TARSKI:def 1;
    then
A2: Hom(c,c)={m} by COMMACAT:4;
    id c in Hom(c,c) by CAT_1:27;
    hence id c = m by A2,TARSKI:def 1;
  end;
