
theorem Th38:
for X be non empty set, Y be set, E be SetSequence of [:X,Y:], x be set
  st E is non-descending
 ex G be SetSequence of X st G is non-descending
   & (for n be Nat holds G.n = Y-section(E.n,x))
proof
   let X be non empty set, Y be set, E be SetSequence of [:X,Y:], x be set;
   assume A1: E is non-descending;
   deffunc F(Nat) = Y-section(E.$1,x);
   consider G be Function of NAT,bool X such that
A2: for n be Element of NAT holds G.n = F(n) from FUNCT_2:sch 4;
   reconsider G as SetSequence of X;
A3:for n be Nat holds G.n = Y-section(E.n,x)
   proof
    let n be Nat;
    n is Element of NAT by ORDINAL1:def 12;
    hence G.n = Y-section(E.n,x) by A2;
   end;
   take G;
   thus G is non-descending by A1,A3,Th34;
   thus thesis by A3;
end;
