let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic AMI-Struct of N
for t, u being State of S
for il being Element of NAT
for e being Element of ObjectKind (IC S)
for I being Element of the Object-Kind of S . il st e = il & u = t +* (((IC S),il) --> (e,I)) holds
( u . il = I & IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) )
let S be non empty stored-program IC-Ins-separated definite realistic AMI-Struct of N; :: thesis: for t, u being State of S
for il being Element of NAT
for e being Element of ObjectKind (IC S)
for I being Element of the Object-Kind of S . il st e = il & u = t +* (((IC S),il) --> (e,I)) holds
( u . il = I & IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) )
let t, u be State of S; :: thesis: for il being Element of NAT
for e being Element of ObjectKind (IC S)
for I being Element of the Object-Kind of S . il st e = il & u = t +* (((IC S),il) --> (e,I)) holds
( u . il = I & IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) )
let il be Element of NAT ; :: thesis: for e being Element of ObjectKind (IC S)
for I being Element of the Object-Kind of S . il st e = il & u = t +* (((IC S),il) --> (e,I)) holds
( u . il = I & IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) )
let e be Element of ObjectKind (IC S); :: thesis: for I being Element of the Object-Kind of S . il st e = il & u = t +* (((IC S),il) --> (e,I)) holds
( u . il = I & IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) )
let I be Element of the Object-Kind of S . il; :: thesis: ( e = il & u = t +* (((IC S),il) --> (e,I)) implies ( u . il = I & IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) ) )
assume that
A1: e = il and
A2: u = t +* (((IC S),il) --> (e,I)) ; :: thesis: ( u . il = I & IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) )
A3: dom (((IC S),il) --> (e,I)) = {(IC S),il} by FUNCT_4:65;
then il in dom (((IC S),il) --> (e,I)) by TARSKI:def 2;
hence u . il = (((IC S),il) --> (e,I)) . il by A2, FUNCT_4:14
.= I by FUNCT_4:66 ;
:: thesis: ( IC u = il & IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) )
IC S in dom (((IC S),il) --> (e,I)) by A3, TARSKI:def 2;
hence IC u = (((IC S),il) --> (e,I)) . (IC S) by A2, FUNCT_4:14
.= il by A1, COMPOS_1:3, FUNCT_4:66 ;
:: thesis: IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S)
thus IC (Following ((ProgramPart u),u)) = (Exec ((u . (IC u)),u)) . (IC S) by COMPOS_1:38; :: thesis: verum