let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being parahalting Program of
for J being Program of
for k being Nat st k <= LifeSpan ((P +* (stop I)),s) holds
Comput ((P +* (stop I)),s,k) = Comput ((P +* (I ';' J)),s,k)
let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of
for J being Program of
for k being Nat st k <= LifeSpan ((P +* (stop I)),s) holds
Comput ((P +* (stop I)),s,k) = Comput ((P +* (I ';' J)),s,k)
let I be parahalting Program of ; :: thesis: for J being Program of
for k being Nat st k <= LifeSpan ((P +* (stop I)),s) holds
Comput ((P +* (stop I)),s,k) = Comput ((P +* (I ';' J)),s,k)
let J be Program of ; :: thesis: for k being Nat st k <= LifeSpan ((P +* (stop I)),s) holds
Comput ((P +* (stop I)),s,k) = Comput ((P +* (I ';' J)),s,k)
let k be Nat; :: thesis: ( k <= LifeSpan ((P +* (stop I)),s) implies Comput ((P +* (stop I)),s,k) = Comput ((P +* (I ';' J)),s,k) )
set spI = stop I;
set P1 = P +* (stop I);
set P2 = P +* (I ';' J);
set n = LifeSpan ((P +* (stop I)),s);
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* (stop I)),s) implies Comput ((P +* (stop I)),s,$1) = Comput ((P +* (I ';' J)),s,$1) );
A1: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let m be Nat; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A2: ( m <= LifeSpan ((P +* (stop I)),s) implies Comput ((P +* (stop I)),s,m) = Comput ((P +* (I ';' J)),s,m) ) ; :: thesis: S₁[m + 1]
A3: Comput ((P +* (I ';' J)),s,(m + 1)) = Following ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m))) by EXTPRO_1:3;
stop I c= P +* (stop I) by FUNCT_4:25;
then A4: IC (Comput ((P +* (stop I)),s,m)) in dom (stop I) by SCMPDS_4:def 6;
A5: Comput ((P +* (stop I)),s,(m + 1)) = Following ((P +* (stop I)),(Comput ((P +* (stop I)),s,m))) by EXTPRO_1:3;
assume A6: m + 1 <= LifeSpan ((P +* (stop I)),s) ; :: thesis: Comput ((P +* (stop I)),s,(m + 1)) = Comput ((P +* (I ';' J)),s,(m + 1))
A7: m < LifeSpan ((P +* (stop I)),s) by A6, NAT_1:13;
then IC (Comput ((P +* (stop I)),s,m)) in dom I by Th12;
then A8: IC (Comput ((P +* (stop I)),s,m)) in dom (I ';' J) by FUNCT_4:12;
A9: IC (Comput ((P +* (stop I)),s,m)) in dom I by A7, Th12;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,m))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),s,m))) by PBOOLE:143
.= (stop I) . (IC (Comput ((P +* (stop I)),s,m))) by A4, FUNCT_4:13
.= I . (IC (Comput ((P +* (stop I)),s,m))) by A9, AFINSQ_1:def 3
.= (I ';' J) . (IC (Comput ((P +* (stop I)),s,m))) by A9, AFINSQ_1:def 3
.= (P +* (I ';' J)) . (IC (Comput ((P +* (stop I)),s,m))) by A8, FUNCT_4:13
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m))) by A6, A2, NAT_1:13, PBOOLE:143 ;
hence Comput ((P +* (stop I)),s,(m + 1)) = Comput ((P +* (I ';' J)),s,(m + 1)) by A2, A6, A5, A3, NAT_1:13; :: thesis: verum

end;

A10: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A10, A1);
hence ( k <= LifeSpan ((P +* (stop I)),s) implies Comput ((P +* (stop I)),s,k) = Comput ((P +* (I ';' J)),s,k) ) ; :: thesis: verum