reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;
reserve f for Morphism of a,b,
        g for Morphism of b,a;
reserve g for Morphism of b,c;

theorem
  f is retraction & g is retraction implies g*f is retraction
proof
  assume
A1: Hom(a,b) <> {} & Hom(b,a) <> {};
  given i being Morphism of b,a such that
A2: f*i = id b;
  assume
A3: Hom(b,c) <> {} & Hom(c,b) <> {};
  given j being Morphism of c,b such that
A4: g*j = id c;
  thus
A5: Hom(a,c) <> {} & Hom(c,a) <> {} by A1,A3,CAT_1:24;
  take i*j;
  thus g*f*(i*j) = g*(f*(i*j)) by A1,A3,A5,CAT_1:25
    .= g*(f*i*j) by A1,A3,CAT_1:25
    .= id c by A2,A3,A4,CAT_1:28;
end;
