let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic COM-Struct of N
for p being FinPartState of S
for k being Element of NAT holds Start-At (((IC p) -' k),S) c= DecIC (p,k)
let S be non empty stored-program IC-Ins-separated definite realistic COM-Struct of N; :: thesis: for p being FinPartState of S
for k being Element of NAT holds Start-At (((IC p) -' k),S) c= DecIC (p,k)
let p be FinPartState of S; :: thesis: for k being Element of NAT holds Start-At (((IC p) -' k),S) c= DecIC (p,k)
let k be Element of NAT ; :: thesis: Start-At (((IC p) -' k),S) c= DecIC (p,k)
A1: IC (DecIC (p,k)) = (IC p) -' k by Th188;
A2: IC in dom (DecIC (p,k)) by Th187;
A3: ( Start-At (((IC p) -' k),S) = {[(IC ),((IC p) -' k)]} & [(IC ),((IC p) -' k)] in DecIC (p,k) ) by A2, A1, FUNCT_1:def 4, FUNCT_4:87;
for x being set st x in Start-At (((IC p) -' k),S) holds
x in DecIC (p,k) by A3, TARSKI:def 1;
hence Start-At (((IC p) -' k),S) c= DecIC (p,k) by TARSKI:def 3; :: thesis: verum