let R be Ring; :: thesis: for k being Nat holds
( k + 1 in SUCC (k,(SCM R)) & ( for j being Nat st j in SUCC (k,(SCM R)) holds
k <= j ) )
let k be Nat; :: thesis: ( k + 1 in SUCC (k,(SCM R)) & ( for j being Nat st j in SUCC (k,(SCM R)) holds
k <= j ) )
A1: SUCC (k,(SCM R)) = {k,(k + 1)} by Th34;
hence k + 1 in SUCC (k,(SCM R)) by TARSKI:def 2; :: thesis: for j being Nat st j in SUCC (k,(SCM R)) holds
k <= j
let j be Nat; :: thesis: ( j in SUCC (k,(SCM R)) implies k <= j )
assume A2: j in SUCC (k,(SCM R)) ; :: thesis: k <= j

per cases ( j = k or j = k + 1 ) by A1, A2, TARSKI:def 2;

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

hence k <= j ; :: thesis: verum

end;

suppose j = k + 1 ; :: thesis: k <= j

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

end;

end;