
theorem Th36:
for X be non empty set, Y be set, x be set, E be SetSequence of [:X,Y:],
 G be SetSequence of X
 st E is non-ascending & (for n be Nat holds G.n = Y-section(E.n,x))
 holds G is non-ascending
proof
   let X be non empty set, Y be set, x be set, E be SetSequence of [:X,Y:],
   G be SetSequence of X;
   assume that
A1: E is non-ascending and
A2: for n be Nat holds G.n = Y-section(E.n,x);
   for n be Nat holds G.(n+1) c= G.n
   proof
    let n be Nat;
    Y-section(E.(n+1),x) c= Y-section(E.n,x) by Th15,A1,KURATO_0:def 3; then
    G.(n+1) c= Y-section(E.n,x) by A2;
    hence G.(n+1) c= G.n by A2;
   end;
   hence G is non-ascending by KURATO_0:def 3;
end;
