let s be State of SCMPDS; :: thesis: for P being Instruction-Sequence of SCMPDS
for I, J being Program of
for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k <= LifeSpan ((P +* (stop I)),(Initialize s)) holds
Comput ((P +* (stop I)),(Initialize s),k) = Comput ((P +* (I ';' J)),(Initialize s),k)
let P be Instruction-Sequence of SCMPDS; :: thesis: for I, J being Program of
for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k <= LifeSpan ((P +* (stop I)),(Initialize s)) holds
Comput ((P +* (stop I)),(Initialize s),k) = Comput ((P +* (I ';' J)),(Initialize s),k)
let I, J be Program of ; :: thesis: for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k <= LifeSpan ((P +* (stop I)),(Initialize s)) holds
Comput ((P +* (stop I)),(Initialize s),k) = Comput ((P +* (I ';' J)),(Initialize s),k)
let k be Nat; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P & k <= LifeSpan ((P +* (stop I)),(Initialize s)) implies Comput ((P +* (stop I)),(Initialize s),k) = Comput ((P +* (I ';' J)),(Initialize s),k) )
set spI = stop I;
set s1 = Initialize s;
set P1 = P +* (stop I);
set s2 = Initialize s;
set P2 = P +* (I ';' J);
set n = LifeSpan ((P +* (stop I)),(Initialize s));
assume that
A1: I is_closed_on s,P and
A2: I is_halting_on s,P ; :: thesis: ( not k <= LifeSpan ((P +* (stop I)),(Initialize s)) or Comput ((P +* (stop I)),(Initialize s),k) = Comput ((P +* (I ';' J)),(Initialize s),k) )
assume A3: k <= LifeSpan ((P +* (stop I)),(Initialize s)) ; :: thesis: Comput ((P +* (stop I)),(Initialize s),k) = Comput ((P +* (I ';' J)),(Initialize s),k)
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* (stop I)),(Initialize s)) implies Comput ((P +* (stop I)),(Initialize s),$1) = Comput ((P +* (I ';' J)),(Initialize s),$1) );
A4: for m being Nat st S₁[m] holds
S₁[m + 1]

proof

let m be Nat; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A5: ( m <= LifeSpan ((P +* (stop I)),(Initialize s)) implies Comput ((P +* (stop I)),(Initialize s),m) = Comput ((P +* (I ';' J)),(Initialize s),m) ) ; :: thesis: S₁[m + 1]
A6: Comput ((P +* (stop I)),(Initialize s),(m + 1)) = Following ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),m))) by EXTPRO_1:3
.= Exec ((CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),m)))),(Comput ((P +* (stop I)),(Initialize s),m))) ;
A7: IC (Comput ((P +* (stop I)),(Initialize s),m)) in dom (stop I) by A1, SCMPDS_6:def 2;
A8: Comput ((P +* (I ';' J)),(Initialize s),(m + 1)) = Following ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(Initialize s),m))) by EXTPRO_1:3
.= Exec ((CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(Initialize s),m)))),(Comput ((P +* (I ';' J)),(Initialize s),m))) ;
assume A9: m + 1 <= LifeSpan ((P +* (stop I)),(Initialize s)) ; :: thesis: Comput ((P +* (stop I)),(Initialize s),(m + 1)) = Comput ((P +* (I ';' J)),(Initialize s),(m + 1))
then m < LifeSpan ((P +* (stop I)),(Initialize s)) by NAT_1:13;
then A10: IC (Comput ((P +* (stop I)),(Initialize s),m)) in dom I by A1, A2, SCMPDS_6:26;
then A11: IC (Comput ((P +* (stop I)),(Initialize s),m)) in dom (I ';' J) by FUNCT_4:12;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),m))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),m))) by PBOOLE:143
.= (stop I) . (IC (Comput ((P +* (stop I)),(Initialize s),m))) by A7, FUNCT_4:13
.= I . (IC (Comput ((P +* (stop I)),(Initialize s),m))) by A10, AFINSQ_1:def 3
.= (I ';' J) . (IC (Comput ((P +* (stop I)),(Initialize s),m))) by A10, AFINSQ_1:def 3
.= (P +* (I ';' J)) . (IC (Comput ((P +* (stop I)),(Initialize s),m))) by A11, FUNCT_4:13
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(Initialize s),m))) by A5, A9, NAT_1:13, PBOOLE:143 ;
hence Comput ((P +* (stop I)),(Initialize s),(m + 1)) = Comput ((P +* (I ';' J)),(Initialize s),(m + 1)) by A5, A9, A8, A6, NAT_1:13; :: thesis: verum

end;

A12: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A12, A4);
hence Comput ((P +* (stop I)),(Initialize s),k) = Comput ((P +* (I ';' J)),(Initialize s),k) by A3; :: thesis: verum