let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite steady-programmed AMI-Struct of N
for P being the Instructions of b₁ -valued ManySortedSet of NAT
for s being State of S
for n being Nat holds ProgramPart s = ProgramPart (Comput (P,s,n))
let S be non empty stored-program IC-Ins-separated definite steady-programmed AMI-Struct of N; :: thesis: for P being the Instructions of S -valued ManySortedSet of NAT
for s being State of S
for n being Nat holds ProgramPart s = ProgramPart (Comput (P,s,n))
let P be the Instructions of S -valued ManySortedSet of NAT ; :: thesis: for s being State of S
for n being Nat holds ProgramPart s = ProgramPart (Comput (P,s,n))
let s be State of S; :: thesis: for n being Nat holds ProgramPart s = ProgramPart (Comput (P,s,n))
defpred S₁[ Nat] means ProgramPart s = ProgramPart (Comput (P,s,$1));

A1: now

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then ProgramPart s = ProgramPart (Following (P,(Comput (P,s,n)))) by L117
.= ProgramPart (Comput (P,s,(n + 1))) by EXTPRO_1:4 ;
hence S₁[n + 1] ; :: thesis: verum

end;

A2: S₁[ 0 ] by EXTPRO_1:3;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A1); :: thesis: verum