
theorem Th33:
for X be set, Y be non empty set, x be set, E be SetSequence of [:X,Y:],
 G be SetSequence of Y
 st E is non-descending & (for n be Nat holds G.n = X-section(E.n,x))
 holds G is non-descending
proof
   let X be set, Y be non empty set, x be set, E be SetSequence of [:X,Y:],
   G be SetSequence of Y;
   assume that
A1: E is non-descending and
A2: for n be Nat holds G.n = X-section(E.n,x);
   for n be Nat holds G.n c= G.(n+1)
   proof
    let n be Nat;
    X-section(E.n,x) c= X-section(E.(n+1),x) by Th14,A1,KURATO_0:def 4; then
    G.n c= X-section(E.(n+1),x) by A2;
    hence G.n c= G.(n+1) by A2;
   end;
   hence G is non-descending by KURATO_0:def 4;
end;
