reconsider ss = s as Element of product the Object-Kind of SCM+FSA by PBOOLE:155;
consider f being Function of NAT,(product the Object-Kind of SCM+FSA) such that
W1: f . 0 = ss and
W2: for i being Nat holds f . (i + 1) = H₂(i,f . i) from NAT_1:sch 12();
take f ; :: thesis: ( f . 0 = s & ( for i being Nat holds f . (i + 1) = Comput ((ProgramPart ((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart ((f . i) +* (I +* (Start-At (0,SCM+FSA))))),((f . i) +* (I +* (Start-At (0,SCM+FSA)))))) + 3)) ) )
thus f . 0 = s by W1; :: thesis: for i being Nat holds f . (i + 1) = Comput ((ProgramPart ((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart ((f . i) +* (I +* (Start-At (0,SCM+FSA))))),((f . i) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))
let i be Nat; :: thesis: f . (i + 1) = Comput ((ProgramPart ((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart ((f . i) +* (I +* (Start-At (0,SCM+FSA))))),((f . i) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))
f . (i + 1) = H₂(i,f . i) by W2;
hence f . (i + 1) = Comput ((ProgramPart ((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),((f . i) +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),((LifeSpan ((ProgramPart ((f . i) +* (I +* (Start-At (0,SCM+FSA))))),((f . i) +* (I +* (Start-At (0,SCM+FSA)))))) + 3)) ; :: thesis: verum