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= IncrIC 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= IncrIC p,k
let p be FinPartState of S; :: thesis: for k being Element of NAT holds Start-At ((IC p) + k),S c= IncrIC p,k
let k be Element of NAT ; :: thesis: Start-At ((IC p) + k),S c= IncrIC p,k
A1: IC (IncrIC p,k) = (IC p) + k by Th19;
A2: IC S in dom (IncrIC p,k) by Th18;
A3: ( Start-At ((IC p) + k),S = {[(IC S),((IC p) + k)]} & [(IC S),((IC p) + k)] in IncrIC p,k ) by A2, A1, FUNCT_1:def 4, FUNCT_4:87;

now

let x be set ; :: thesis: ( x in Start-At ((IC p) + k),S implies x in IncrIC p,k )
assume x in Start-At ((IC p) + k),S ; :: thesis: x in IncrIC p,k
hence x in IncrIC p,k by A3, TARSKI:def 1; :: thesis: verum

end;

hence Start-At ((IC p) + k),S c= IncrIC p,k by TARSKI:def 3; :: thesis: verum