let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program halting IC-Ins-separated steady-programmed definite realistic standard AMI-Struct of N holds (il. S,0 ) .--> (halt S) is closed
let S be non empty stored-program halting IC-Ins-separated steady-programmed definite realistic standard AMI-Struct of N; :: thesis: (il. S,0 ) .--> (halt S) is closed
reconsider F = (il. S,0 ) .--> (halt S) as NAT -defined FinPartState of ;
let l be Element of NAT ; :: according to AMISTD_1:def 17 :: thesis: ( l in dom ((il. S,0 ) .--> (halt S)) implies NIC (pi ((il. S,0 ) .--> (halt S)),l),l c= dom ((il. S,0 ) .--> (halt S)) )
assume A1: l in dom ((il. S,0 ) .--> (halt S)) ; :: thesis: NIC (pi ((il. S,0 ) .--> (halt S)),l),l c= dom ((il. S,0 ) .--> (halt S))
A2: dom F = {(il. S,0 )} by FUNCOP_1:19;
then A3: l = il. S,0 by A1, TARSKI:def 1;
pi F,l = F . l by A1, AMI_1:def 47
.= halt S by A3, FUNCOP_1:87 ;
hence NIC (pi ((il. S,0 ) .--> (halt S)),l),l c= dom ((il. S,0 ) .--> (halt S)) by A2, A3, Th15; :: thesis: verum