let R be Ring; :: thesis: for a, c being Data-Location of R
for i1 being Nat
for s being State of (SCM R) holds
( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )
let a, c be Data-Location of R; :: thesis: for i1 being Nat
for s being State of (SCM R) holds
( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )
let i1 be Nat; :: thesis: for s being State of (SCM R) holds
( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )
let s be State of (SCM R); :: thesis: ( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )
A1: the_Values_of (SCM R) = (SCM-VAL R) * SCM-OK by Lm1;
reconsider S = s as SCM-State of R by A1, CARD_3:107;
reconsider I = a =0_goto i1 as Element of SCM-Instr R by Def1;
reconsider i = 7 as Element of Segm 8 by NAT_1:44;
A2: ( a is Element of Data-Locations & i1 is Element of NAT ) by Th1, ORDINAL1:def 12;
A3: Exec ((a =0_goto i1),s) = SCM-Exec-Res (I,S) by Th10
.= SCM-Chg (S,(IFEQ ((S . (I cond_address)),(0. R),(I cjump_address),((IC S) + 1)))) by A2, AMI_3:27, SCMRING1:def 14 ;
A4: I = [i,<*i1*>,<*a*>] ;
thus ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = i1 ) :: thesis: ( ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )

proof

assume s . a = 0. R ; :: thesis: (Exec ((a =0_goto i1),s)) . (IC ) = i1
then A5: S . (I cond_address) = 0. R by A4, A2, AMI_3:27, SCMRINGI:3;
thus (Exec ((a =0_goto i1),s)) . (IC ) = (Exec ((a =0_goto i1),s)) . NAT by Def1
.= IFEQ ((S . (I cond_address)),(0. R),(I cjump_address),((IC S) + 1)) by A3, SCMRING1:7
.= I cjump_address by A5, FUNCOP_1:def 8
.= i1 by A4, A2, AMI_3:27, SCMRINGI:3 ; :: thesis: verum

end;

A6: IC s = IC S by Def1;
thus ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . (IC ) = (IC s) + 1 ) :: thesis: (Exec ((a =0_goto i1),s)) . c = s . c

proof

assume s . a <> 0. R ; :: thesis: (Exec ((a =0_goto i1),s)) . (IC ) = (IC s) + 1
then A7: S . (I cond_address) <> 0. R by A4, A2, AMI_3:27, SCMRINGI:3;
thus (Exec ((a =0_goto i1),s)) . (IC ) = (Exec ((a =0_goto i1),s)) . NAT by Def1
.= IFEQ ((S . (I cond_address)),(0. R),(I cjump_address),((IC S) + 1)) by A3, SCMRING1:7
.= (IC s) + 1 by A6, A7, FUNCOP_1:def 8 ; :: thesis: verum

end;

c is Element of Data-Locations by Th1;
hence (Exec ((a =0_goto i1),s)) . c = s . c by A3, AMI_3:27, SCMRING1:8; :: thesis: verum