reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;
reserve e1,e2 for ExtReal;

theorem Th7:
  (seq is increasing iff for n,m be Nat st m<n holds seq
.m<seq.n) & (seq is decreasing iff for n,m be Nat st m<n holds seq.n
  <seq.m) & (seq is non-decreasing iff for n,m be Nat st m<=n holds
  seq.m<=seq.n) & (seq is non-increasing iff
   for n,m be Nat st m<=n holds seq.n<=seq.m)
proof

A1:  dom seq = NAT by FUNCT_2:def 1;
 thus seq is increasing implies for n,m be Nat st m<n holds seq.m<seq.n
  proof assume
A2: seq is increasing;
   let n,m be Nat;
    n in NAT & m in NAT by ORDINAL1:def 12;
   hence thesis by A2,A1;
  end;
 thus (for n,m be Nat st m<n holds seq.m<seq.n) implies seq is increasing;
 thus seq is decreasing implies for n,m be Nat st m<n holds seq.m>seq.n
  proof assume
A3: seq is decreasing;
   let n,m be Nat;
    n in NAT & m in NAT by ORDINAL1:def 12;
   hence thesis by A3,A1;
  end;
  thus (for n,m be Nat st m<n holds seq.n <seq.m) implies seq is decreasing;
 thus seq is non-decreasing implies for n,m be Nat st m<=n holds seq.m<=seq.n
  proof assume
A4: seq is non-decreasing;
   let n,m be Nat;
    n in NAT & m in NAT by ORDINAL1:def 12;
   hence thesis by A4,A1;
  end;
 thus (for n,m be Nat st m<=n holds seq.m<=seq.n)
  implies seq is non-decreasing;
 thus seq is non-increasing implies for n,m be Nat st m<=n holds seq.m>=seq.n
  proof assume
A5: seq is non-increasing;
   let n,m be Nat;
    n in NAT & m in NAT by ORDINAL1:def 12;
   hence thesis by A5,A1;
  end;
    assume
A6: for n,m be Nat st m<=n holds seq.n<=seq.m;
    let e1,e2;
    thus thesis by A6;
end;
