let k be Element of NAT ; :: thesis: ( k + 1 in SUCC (k,SCM) & ( for j being Element of NAT st j in SUCC (k,SCM) holds
k <= j ) )
reconsider fk = k as Element of NAT ;
A1: SUCC (k,SCM) = {k,(succ fk)} by Th54;
hence k + 1 in SUCC (k,SCM) by TARSKI:def 2; :: thesis: for j being Element of NAT st j in SUCC (k,SCM) holds
k <= j
let j be Element of NAT ; :: thesis: ( j in SUCC (k,SCM) implies k <= j )
assume A2: j in SUCC (k,SCM) ; :: thesis: k <= j
reconsider fk = k as Element of NAT ;

per cases ( j = k or j = succ fk ) by A1, A2, TARSKI:def 2;

suppose j = k ; :: thesis: k <= j

hence k <= j ; :: thesis: verum

end;

suppose j = succ fk ; :: thesis: k <= j

hence k <= j by NAT_1:11; :: thesis: verum

end;

end;