
theorem
for X,Y be non empty set, F be disjoint_valued SetSequence of [:X,Y:],
  p be set holds
  ( ex Fy be disjoint_valued SetSequence of X st
     (for n be Nat holds Fy.n = Y-section(F.n,p)) )
& ( ex Fx be disjoint_valued SetSequence of Y st
     (for n be Nat holds Fx.n = X-section(F.n,p)) )
proof
   let X,Y be non empty set, F be disjoint_valued SetSequence of [:X,Y:],
    p be set;
   thus ex Fy be disjoint_valued SetSequence of X st
     (for n be Nat holds Fy.n = Y-section(F.n,p))
   proof
    deffunc f(Nat) = Y-section(F.$1,p);
    consider Fy be SetSequence of X such that
A1:  for n be Element of NAT holds Fy.n = f(n) from FUNCT_2:sch 4;
    now let n,m be Nat;
A2:  n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;
     assume n <> m; then
     F.n misses F.m by PROB_3:def 4; then
     Y-section(F.n,p) misses Y-section(F.m,p) by Th29; then
     Fy.n misses Y-section(F.m,p) by A1,A2;
     hence Fy.n misses Fy.m by A1,A2;
    end; then
    reconsider Fy as disjoint_valued SetSequence of X by PROB_3:def 4;
    take Fy;
    let n be Nat;
    n is Element of NAT by ORDINAL1:def 12;
    hence Fy.n = Y-section(F.n,p) by A1;
   end;
    deffunc f(Nat) = X-section(F.$1,p);
    consider Fx be SetSequence of Y such that
A3:  for n be Element of NAT holds Fx.n = f(n) from FUNCT_2:sch 4;
    now let n,m be Nat;
A4:  n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;
     assume n <> m; then
     F.n misses F.m by PROB_3:def 4; then
     X-section(F.n,p) misses X-section(F.m,p) by Th29; then
     Fx.n misses X-section(F.m,p) by A3,A4;
     hence Fx.n misses Fx.m by A3,A4;
    end; then
    reconsider Fx as disjoint_valued SetSequence of Y by PROB_3:def 4;
    take Fx;
    let n be Nat;
     n is Element of NAT by ORDINAL1:def 12;
     hence Fx.n = X-section(F.n,p) by A3;
end;
