
theorem Th58:
for X1,X2,Y be non empty set, F be Functional_Sequence of [:X1,X2:],Y,
  x be Element of X1, y be Element of X2 st F is with_the_same_dom
  holds ProjPMap1(F,x) is with_the_same_dom
      & ProjPMap2(F,y) is with_the_same_dom
proof
    let X1,X2,Y be non empty set, F be Functional_Sequence of [:X1,X2:],Y,
     x1 be Element of X1, x2 be Element of X2;
    assume A1: F is with_the_same_dom;

    now let m,n be Nat;
     dom(ProjPMap1(F,x1).m) = dom(ProjPMap1(F.m,x1)) by Def5
      .= X-section(dom(F.m),x1) by Def3
      .= X-section(dom(F.n),x1) by A1,MESFUNC8:def 2
      .= dom(ProjPMap1(F.n,x1)) by Def3;
     hence dom(ProjPMap1(F,x1).m) = dom(ProjPMap1(F,x1).n) by Def5;
    end;
    hence ProjPMap1(F,x1) is with_the_same_dom by MESFUNC8:def 2;

    now let m,n be Nat;
     dom(ProjPMap2(F,x2).m) = dom(ProjPMap2(F.m,x2)) by Def6
      .= Y-section(dom(F.m),x2) by Def4
      .= Y-section(dom(F.n),x2) by A1,MESFUNC8:def 2
      .= dom(ProjPMap2(F.n,x2)) by Def4;
     hence dom(ProjPMap2(F,x2).m) = dom(ProjPMap2(F,x2).n) by Def6;
    end;
    hence ProjPMap2(F,x2) is with_the_same_dom by MESFUNC8:def 2;
end;
