let s be State of ; :: thesis: for I being Program of
for l being Instruction-Location of SCMPDS holds DataPart s = DataPart (s +* (I +* (Start-At l)))
let I be Program of ; :: thesis: for l being Instruction-Location of SCMPDS holds DataPart s = DataPart (s +* (I +* (Start-At l)))
let l be Instruction-Location of SCMPDS ; :: thesis: DataPart s = DataPart (s +* (I +* (Start-At l)))

now

let x be set ; :: thesis: ( x in dom (I +* (Start-At l)) implies not b₁ in SCM-Data-Loc )
assume x in dom (I +* (Start-At l)) ; :: thesis: not b₁ in SCM-Data-Loc
then x in (dom I) \/ (dom (Start-At l)) by FUNCT_4:def 1;
then ( x in dom I or x in dom (Start-At l) ) by XBOOLE_0:def 3;
then A1: ( x in dom I or x in {(IC SCMPDS )} ) by FUNCOP_1:19;

per cases ( x in dom I or x = IC SCMPDS ) by A1, TARSKI:def 1;

suppose A2: x in dom I ; :: thesis: not b₁ in SCM-Data-Loc

dom I c= NAT by RELAT_1:def 18;
hence not x in SCM-Data-Loc by A2, AMI_2:29, XBOOLE_0:3; :: thesis: verum

end;

suppose x = IC SCMPDS ; :: thesis: not b₁ in SCM-Data-Loc

hence not x in SCM-Data-Loc by SCMPDS_3:6; :: thesis: verum

end;

end;

end;

then dom (I +* (Start-At l)) misses SCM-Data-Loc by XBOOLE_0:3;
hence DataPart s = DataPart (s +* (I +* (Start-At l))) by FUNCT_4:94, SCMPDS_2:100; :: thesis: verum