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;

theorem Th7:
  cods((x1,x2)-->(p1,p2)) = (x1,x2)-->(cod p1,cod p2)
proof
  set F = (x1,x2)-->(p1,p2), f = x1 .--> p1, g = x2 .--> p2, F9 = (x1,x2)-->(
  cod p1,cod p2), f9 = x1.-->(cod p1), g9 = x2.-->(cod p2);
A1: dom g = {x2} by FUNCOP_1:13;
A2: dom g9 = {x2} & F9 = f9 +* g9 by FUNCOP_1:13,FUNCT_4:def 4;
A3: F = f +* g by FUNCT_4:def 4;
A4: dom f = {x1} by FUNCOP_1:13;
  now
    let x;
    assume
A5: x in {x1,x2};
    then
A6: x in dom F by FUNCT_4:62;
    now
      per cases by A3,A6,FUNCT_4:12;
      case
A7:     x in dom f & not x in dom g;
        then F.x = f.x by A3,FUNCT_4:11;
        then
A8:     F.x = p1 by A4,A7,FUNCOP_1:7;
        F9.x = f9.x by A1,A2,A7,FUNCT_4:11;
        then F9.x = cod p1 by A4,A7,FUNCOP_1:7;
        hence F/.x = p1 & F9/.x = cod p1 by A5,A8,FUNCT_2:def 13;
      end;
      case
A9:     x in dom g;
        then F.x = g.x by A3,FUNCT_4:13;
        then
A10:    F.x = p2 by A1,A9,FUNCOP_1:7;
        F9.x = g9.x by A1,A2,A9,FUNCT_4:13;
        then F9.x = cod p2 by A1,A9,FUNCOP_1:7;
        hence F/.x = p2 & F9/.x = cod p2 by A5,A10,FUNCT_2:def 13;
      end;
    end;
    hence (cods F)/.x = F9/.x by A5,Def2;
  end;
  hence thesis by Th1;
end;
