
theorem Th8:
  for x being object, C being CategoryStr st
  the carrier of C = {x} & the composition of C = {[[x,x],x]}
  holds C is non empty category
  proof
    let x be object;
    let C be CategoryStr;
    assume
A1: the carrier of C = {x};
    assume
A2: the composition of C = {[[x,x],x]};
A3: the carrier of C = the carrier of DiscreteCat({x}) by A1,CAT_6:def 16;
    for y being object holds y in the composition of DiscreteCat({x})
    iff y in {[[x,x],x]}
    proof
      let y be object;
      hereby
        assume y in the composition of DiscreteCat({x});
        then consider x1,x2 be object such that
A4:     y = [x1,x2] & x1 in [: the carrier of DiscreteCat({x}),
        the carrier of DiscreteCat({x}):] &
        x2 in the carrier of DiscreteCat({x}) by RELSET_1:2;
A5:     x1 in [:{x},{x}:] & x2 = x by A3,A4,A1,TARSKI:def 1;
        x1 in {[x,x]} by ZFMISC_1:29,A3,A4,A1;
        then x1 = [x,x] by TARSKI:def 1;
        hence y in {[[x,x],x]} by A5,A4,TARSKI:def 1;
      end;
      assume
A6:  y in {[[x,x],x]};
      x in the carrier of DiscreteCat({x}) by A3,A1,TARSKI:def 1;
      then reconsider f = x as morphism of DiscreteCat({x}) by CAT_6:def 1;
A7:  y = [[f,f],f] by A6,TARSKI:def 1;
A8:   DiscreteCat({x}) is non empty by CAT_6:def 16;
A9:  f is identity by CAT_6:def 15;
A10:   f |> f by A8,CAT_6:24,CAT_6:def 15;
      then
A11:   [f,f] in dom the composition of DiscreteCat({x}) by CAT_6:def 2;
      f = f (*) f by A9,A10,Th4;
      then (the composition of DiscreteCat({x})).(f,f) = f by A10,CAT_6:def 3;
      then (the composition of DiscreteCat({x})).[f,f] = f by BINOP_1:def 1;
      hence y in the composition of DiscreteCat({x}) by A7,A11,FUNCT_1:1;
    end;
    then the composition of DiscreteCat({x}) = {[[x,x],x]} by TARSKI:2;
    hence thesis by A3,A2,CAT_6:14,15,20;
  end;
