let F be NAT -defined the Instructions of SCM -valued total Function; :: thesis: for k, n being Element of NAT
for s being State of SCM
for a being Data-Location
for il being Element of NAT st IC (Comput (F,s,k)) = n & F . n = a >0_goto il holds
( ( (Comput (F,s,k)) . a > 0 implies IC (Comput (F,s,(k + 1))) = il ) & ( (Comput (F,s,k)) . a <= 0 implies IC (Comput (F,s,(k + 1))) = n + 1 ) & ( for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d ) )
let k, n be Element of NAT ; :: thesis: for s being State of SCM
for a being Data-Location
for il being Element of NAT st IC (Comput (F,s,k)) = n & F . n = a >0_goto il holds
( ( (Comput (F,s,k)) . a > 0 implies IC (Comput (F,s,(k + 1))) = il ) & ( (Comput (F,s,k)) . a <= 0 implies IC (Comput (F,s,(k + 1))) = n + 1 ) & ( for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d ) )
let s be State of SCM; :: thesis: for a being Data-Location
for il being Element of NAT st IC (Comput (F,s,k)) = n & F . n = a >0_goto il holds
( ( (Comput (F,s,k)) . a > 0 implies IC (Comput (F,s,(k + 1))) = il ) & ( (Comput (F,s,k)) . a <= 0 implies IC (Comput (F,s,(k + 1))) = n + 1 ) & ( for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d ) )
let a be Data-Location ; :: thesis: for il being Element of NAT st IC (Comput (F,s,k)) = n & F . n = a >0_goto il holds
( ( (Comput (F,s,k)) . a > 0 implies IC (Comput (F,s,(k + 1))) = il ) & ( (Comput (F,s,k)) . a <= 0 implies IC (Comput (F,s,(k + 1))) = n + 1 ) & ( for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d ) )
let il be Element of NAT ; :: thesis: ( IC (Comput (F,s,k)) = n & F . n = a >0_goto il implies ( ( (Comput (F,s,k)) . a > 0 implies IC (Comput (F,s,(k + 1))) = il ) & ( (Comput (F,s,k)) . a <= 0 implies IC (Comput (F,s,(k + 1))) = n + 1 ) & ( for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d ) ) )
assume that
A1: IC (Comput (F,s,k)) = n and
A2: F . n = a >0_goto il ; :: thesis: ( ( (Comput (F,s,k)) . a > 0 implies IC (Comput (F,s,(k + 1))) = il ) & ( (Comput (F,s,k)) . a <= 0 implies IC (Comput (F,s,(k + 1))) = n + 1 ) & ( for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d ) )
set csk1 = Comput (F,s,(k + 1));
set csk = Comput (F,s,k);
A3: dom F = NAT by PARTFUN1:def 2;
A4: Comput (F,s,(k + 1)) = Exec ((CurInstr (F,(Comput (F,s,k)))),(Comput (F,s,k))) by Lm3
.= Exec ((a >0_goto il),(Comput (F,s,k))) by A1, A2, A3, PARTFUN1:def 6 ;
hence ( (Comput (F,s,k)) . a > 0 implies IC (Comput (F,s,(k + 1))) = il ) by AMI_3:9; :: thesis: ( ( (Comput (F,s,k)) . a <= 0 implies IC (Comput (F,s,(k + 1))) = n + 1 ) & ( for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d ) )

hereby :: thesis: for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d

assume (Comput (F,s,k)) . a <= 0 ; :: thesis: IC (Comput (F,s,(k + 1))) = n + 1
hence IC (Comput (F,s,(k + 1))) = succ n by A1, A4, AMI_3:9
.= n + 1 ;
:: thesis: verum

end;

thus for d being Data-Location holds (Comput (F,s,(k + 1))) . d = (Comput (F,s,k)) . d by A4, AMI_3:9; :: thesis: verum