let p be FinPartState of SCM+FSA ; :: thesis: for k being Element of NAT holds ProgramPart (Relocated p,k) = IncAddr (Shift (ProgramPart p),k),k
let k be Element of NAT ; :: thesis: ProgramPart (Relocated p,k) = IncAddr (Shift (ProgramPart p),k),k
set X = (Start-At ((IC p) + k),SCM+FSA ) | NAT ;
consider x being Element of dom ((Start-At ((IC p) + k),SCM+FSA ) | NAT );

A1: now

assume (Start-At ((IC p) + k),SCM+FSA ) | NAT <> {} ; :: thesis: contradiction
then dom ((Start-At ((IC p) + k),SCM+FSA ) | NAT ) <> {} ;
then x in dom ((Start-At ((IC p) + k),SCM+FSA ) | NAT ) ;
then A2: x in (dom (Start-At ((IC p) + k),SCM+FSA )) /\ NAT by RELAT_1:90;
then reconsider x = x as Element of NAT by XBOOLE_0:def 4;
x in dom (Start-At ((IC p) + k),SCM+FSA ) by A2, XBOOLE_0:def 4;
then x in {(IC SCM+FSA )} by FUNCOP_1:19;
then x = IC SCM+FSA by TARSKI:def 1;
hence contradiction by AMI_1:48; :: thesis: verum

end;

reconsider kk = (Start-At ((IC p) + k),SCM+FSA ) | NAT as Function ;
A3: dom (IncAddr (Shift (ProgramPart p),k),k) c= NAT by RELAT_1:def 18;
reconsider rr = ((IncAddr (Shift (ProgramPart p),k),k) +* (DataPart p)) | NAT as Function ;
A4: dom (DataPart p) c= Int-Locations \/ FinSeq-Locations by RELAT_1:87, SCMFSA_2:127;
thus ProgramPart (Relocated p,k) = ((Start-At ((IC p) + k),SCM+FSA ) +* ((IncAddr (Shift (ProgramPart p),k),k) +* (DataPart p))) | NAT by FUNCT_4:15
.= kk +* rr by FUNCT_4:75
.= ((IncAddr (Shift (ProgramPart p),k),k) +* (DataPart p)) | NAT by A1, FUNCT_4:21
.= IncAddr (Shift (ProgramPart p),k),k by A3, A4, Lm1, FUNCT_4:81 ; :: thesis: verum