defpred S₁[ Nat] means CurInstr p,(Comput p,s,$1) = halt S;
consider k being Nat such that
IC (Comput p,s,k) in dom p and
W: CurInstr p,(Comput p,s,k) = halt S by A1, Def20;
A2: ex k being Nat st S₁[k] by W;
consider k being Nat such that
A3: ( S₁[k] & ( for n being Nat st S₁[n] holds
k <= n ) ) from NAT_1:sch 5(A2);
reconsider k = k as Element of NAT by ORDINAL1:def 13;
take k ; :: thesis: ( CurInstr p,(Comput p,s,k) = halt S & ( for k being Element of NAT st CurInstr p,(Comput p,s,k) = halt S holds
k <= k ) )
thus ( CurInstr p,(Comput p,s,k) = halt S & ( for k being Element of NAT st CurInstr p,(Comput p,s,k) = halt S holds
k <= k ) ) by A3; :: thesis: verum