let seq be Real_Sequence; :: thesis: ( ( ( for n being Nat holds seq . n < seq . (n + 1) ) implies for n, k being Nat holds seq . n < seq . ((n + 1) + k) ) & ( ( for n, k being Nat holds seq . n < seq . ((n + 1) + k) ) implies for n, m being Nat st n < m holds
seq . n < seq . m ) & ( ( for n, m being Nat st n < m holds
seq . n < seq . m ) implies for n being Nat holds seq . n < seq . (n + 1) ) )
thus ( ( for n being Nat holds seq . n < seq . (n + 1) ) implies for n, k being Nat holds seq . n < seq . ((n + 1) + k) ) :: thesis: ( ( ( for n, k being Nat holds seq . n < seq . ((n + 1) + k) ) implies for n, m being Nat st n < m holds
seq . n < seq . m ) & ( ( for n, m being Nat st n < m holds
seq . n < seq . m ) implies for n being Nat holds seq . n < seq . (n + 1) ) )

proof

assume A1: for n being Nat holds seq . n < seq . (n + 1) ; :: thesis: for n, k being Nat holds seq . n < seq . ((n + 1) + k)
let n be Nat; :: thesis: for k being Nat holds seq . n < seq . ((n + 1) + k)
defpred S₁[ Nat] means seq . n < seq . ((n + 1) + $1);

A2: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
seq . ((n + 1) + k) < seq . (((n + 1) + k) + 1) by A1;
hence S₁[k + 1] by A3, XXREAL_0:2; :: thesis: verum

end;

A4: S₁[ 0 ] by A1;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A2); :: thesis: verum

end;

thus ( ( for n, k being Nat holds seq . n < seq . ((n + 1) + k) ) implies for n, m being Nat st n < m holds
seq . n < seq . m ) :: thesis: ( ( for n, m being Nat st n < m holds
seq . n < seq . m ) implies for n being Nat holds seq . n < seq . (n + 1) )

proof

assume A5: for n, k being Nat holds seq . n < seq . ((n + 1) + k) ; :: thesis: for n, m being Nat st n < m holds
seq . n < seq . m
let n, m be Nat; :: thesis: ( n < m implies seq . n < seq . m )
assume n < m ; :: thesis: seq . n < seq . m
then ex k being Nat st m = (n + 1) + k by Lm1;
hence seq . n < seq . m by A5; :: thesis: verum

end;

assume A6: for n, m being Nat st n < m holds
seq . n < seq . m ; :: thesis: for n being Nat holds seq . n < seq . (n + 1)
let n be Nat; :: thesis: seq . n < seq . (n + 1)
n < n + 1 by NAT_1:13;
hence seq . n < seq . (n + 1) by A6; :: thesis: verum