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

proof

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

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 + k) + 1) <= seq . (n + k) by A1;
hence S₁[k + 1] by A3, XXREAL_0:2; :: thesis: verum

end;

A4: S₁[ 0 ] ;
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 + k) <= seq . n ) implies for n, m being Nat st n <= m holds
seq . m <= seq . n ) :: thesis: ( ( for n, m being Nat st n <= m holds
seq . m <= seq . n ) implies for n being Nat holds seq . (n + 1) <= seq . n )

proof

assume A5: for n, k being Nat holds seq . (n + k) <= seq . n ; :: thesis: for n, m being Nat st n <= m holds
seq . m <= seq . n
let n, m be Nat; :: thesis: ( n <= m implies seq . m <= seq . n )
assume n <= m ; :: thesis: seq . m <= seq . n
then consider k1 being Nat such that
A6: m = n + k1 by NAT_1:10;
thus seq . m <= seq . n by A5, A6; :: thesis: verum

end;

assume A7: for n, m being Nat st n <= m holds
seq . m <= seq . n ; :: thesis: for n being Nat holds seq . (n + 1) <= seq . n
let n be Nat; :: thesis: seq . (n + 1) <= seq . n
n <= n + 1 by NAT_1:13;
hence seq . (n + 1) <= seq . n by A7; :: thesis: verum