
theorem Th29:
  for C being Category, a,b,c being Object of C for x being set
holds x in Args(compsym(a,b,c), MSAlg C) iff ex g,f being Morphism of C st x =
  <*g,f*> & dom f = a & cod f = b & dom g = b & cod g = c
proof
  let C be Category, a,b,c be Object of C, x be set;
  set A = MSAlg C;
  for a,b being Object of C holds (the Sorts of A).homsym(a,b) = Hom(a,b )
  by Def13;
  then
A1: Args(compsym(a,b,c), A) = product <*Hom(b,c),Hom(a,b)*> by Lm5;
  hereby
    assume x in Args(compsym(a,b,c), A);
    then consider g,f being object such that
A2: g in Hom(b,c) & f in Hom(a,b) and
A3: x = <*g,f*> by A1,FINSEQ_3:124;
    reconsider g,f as Morphism of C by A2;
    take g,f;
    thus x = <*g,f*> by A3;
    thus dom f = a & cod f = b & dom g = b & cod g = c by A2,CAT_1:1;
  end;
  given g,f being Morphism of C such that
A4: x = <*g,f*> and
A5: dom f = a & cod f = b & dom g = b & cod g = c;
  f in Hom(a,b) & g in Hom(b,c) by A5;
  hence thesis by A1,A4,FINSEQ_3:124;
end;
