let E be set ; :: thesis: for A being Subset of (E ^omega)
for m, n being Nat st m <= n holds
A |^ (m,n) = (A |^ m) \/ (A |^ ((m + 1),n))
let A be Subset of (E ^omega); :: thesis: for m, n being Nat st m <= n holds
A |^ (m,n) = (A |^ m) \/ (A |^ ((m + 1),n))
let m, n be Nat; :: thesis: ( m <= n implies A |^ (m,n) = (A |^ m) \/ (A |^ ((m + 1),n)) )
assume A1: m <= n ; :: thesis: A |^ (m,n) = (A |^ m) \/ (A |^ ((m + 1),n))

per cases ( m < n or m = n ) by A1, XXREAL_0:1;

suppose m < n ; :: thesis: A |^ (m,n) = (A |^ m) \/ (A |^ ((m + 1),n))

then A |^ (m,n) = (A |^ (m,m)) \/ (A |^ ((m + 1),n)) by Th25;
hence A |^ (m,n) = (A |^ m) \/ (A |^ ((m + 1),n)) by Th22; :: thesis: verum

end;

suppose A2: m = n ; :: thesis: A |^ (m,n) = (A |^ m) \/ (A |^ ((m + 1),n))

then A3: m + 1 > n by NAT_1:13;
thus A |^ (m,n) = (A |^ m) \/ {} by A2, Th22
.= (A |^ m) \/ (A |^ ((m + 1),n)) by A3, Th21 ; :: thesis: verum

end;

end;