let f be Function; :: thesis: ( ( for i being Element of NAT holds f . (i + 1) c= f . i ) implies for i, j being Element of NAT st i <= j holds
f . j c= f . i )
assume A1: for i being Element of NAT holds f . (i + 1) c= f . i ; :: thesis: for i, j being Element of NAT st i <= j holds
f . j c= f . i
let i, j be Element of NAT ; :: thesis: ( i <= j implies f . j c= f . i )
assume A2: i <= j ; :: thesis: f . j c= f . i
defpred S₁[ Element of NAT ] means ( i + $1 <= j implies f . (i + $1) c= f . i );
A3: S₁[ 0 ] ;

A4: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
A5: (i + k) + 1 = i + (k + 1) ;
assume A6: S₁[k] ; :: thesis: S₁[k + 1]
f . (i + (k + 1)) c= f . (i + k) by A1, A5;
hence S₁[k + 1] by A5, A6, NAT_1:13, XBOOLE_1:1; :: thesis: verum

end;

A7: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A3, A4);
consider k being Nat such that
A8: i + k = j by A2, NAT_1:10;
k in NAT by ORDINAL1:def 13;
hence f . j c= f . i by A7, A8; :: thesis: verum