
theorem id1:
for R being Ring,
    C being ascending Chain of Ideals(R)
holds union the set of all C.i where i is Nat is Ideal of R
proof
let R be Ring, F be ascending Chain of Ideals(R);
set G = the set of all F.i where i is Nat;
set T = union G;
M: F.0 in G;
F.0 in Ideals R;
then consider I being Ideal of R such that H: I = F.0;
set x = the Element of I;
L: ex Y being set st x in Y & Y in G by M,H;
now let x be object;
  assume x in T; then
  consider x1 being set such that H1: x in x1 & x1 in G by TARSKI:def 4;
  consider i being Nat such that H2: x1 = F.i by H1;
  F.i in Ideals R;
  hence x in the carrier of R by H1,H2;
  end;
then T c= the carrier of R;
then reconsider T as non empty Subset of R by L,TARSKI:def 4;
now let x,y be Element of R;
  assume H0: x in T & y in T; then
  consider x1 being set such that H1: x in x1 & x1 in G by TARSKI:def 4;
  consider i being Nat such that H2: x1 = F.i by H1;
  F.i in Ideals R;
  then consider Ix being Ideal of R such that H5: Ix = F.i;
  consider y1 being set such that H3: y in y1 & y1 in G by H0,TARSKI:def 4;
  consider j being Nat such that H4: y1 = F.j by H3;
  F.j in Ideals R;
  then consider Iy being Ideal of R such that H6: Iy = F.j;
  now per cases;
  suppose i <= j;
    then x in Iy by H6,H2,H1,ch1;
    hence ex Y being set st x+y in Y & Y in G by H3,H4,H6,IDEAL_1:def 1;
    end;
  suppose j <= i;
    then y in Ix by H5,H3,H4,ch1;
    hence ex Y being set st x+y in Y & Y in G by H2,H1,H5,IDEAL_1:def 1;
    end;
  end;
  hence x + y in T by TARSKI:def 4;
  end;
then A: T is add-closed;
now let p,x be Element of R;
  assume x in T; then
  consider x1 being set such that H1: x in x1 & x1 in G by TARSKI:def 4;
  consider i being Nat such that H2: x1 = F.i by H1;
  F.i in Ideals R;
  then consider Ix being Ideal of R such that H5: Ix = F.i;
  p * x in Ix by H1,H2,H5,IDEAL_1:def 2;
  hence p * x in T by H5,H2,H1,TARSKI:def 4;
  end;
then B: T is left-ideal;
now let p,x be Element of R;
  assume x in T; then
  consider x1 being set such that H1: x in x1 & x1 in G by TARSKI:def 4;
  consider i being Nat such that H2: x1 = F.i by H1;
  F.i in Ideals R;
  then consider Ix being Ideal of R such that H5: Ix = F.i;
  x * p in Ix by H1,H2,H5,IDEAL_1:def 3;
  hence x * p in T by H1,H2,H5,TARSKI:def 4;
  end;
then T is right-ideal;
hence thesis by A,B;
end;
