defpred S₁[ Nat] means CurInstr (Computation s,$1) = halt S;
consider k being Element of NAT such that
IC (Computation s,k) in dom (ProgramPart s) and
W: (ProgramPart s) . (IC (Computation s,k)) = halt S by A1, Def20;
(ProgramPart s) . (IC (Computation s,k)) = CurInstr (Computation s,k) by LmX;
then 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 (Computation s,k) = halt S & ( for k being Element of NAT st CurInstr (Computation s,k) = halt S holds
k <= k ) )
thus ( CurInstr (Computation s,k) = halt S & ( for k being Element of NAT st CurInstr (Computation s,k) = halt S holds
k <= k ) ) by A3; :: thesis: verum