reserve j, k, m, n for Nat,
  a,b for Int_position,
  k1,k2 for Integer;
reserve P,P1,P2 for Instruction-Sequence of SCMPDS;

theorem Th7:
  for s1,s2 being State of SCMPDS,k1,k2,m be Integer st IC s1= IC
  s2 & k1 <> k2 & m=IC s1 & m+k1 >= 0 & m+k2 >= 0 holds ICplusConst(s1,k1) <>
  ICplusConst(s2,k2)
proof
  let s1,s2 be State of SCMPDS,k1,k2,m be Integer;
  assume that
A1: IC s1 = IC s2 and
A2: k1<>k2 and
A3: m=IC s1 and
A4: m+k1 >= 0 and
A5: m+k2 >= 0;
  ex i being Element of NAT st i = IC s1 & ICplusConst(s1, k1)=|.i+k1.| by
SCMPDS_2:def 18;
  then
A6: ICplusConst(s1,k1)=m+k1 by A3,A4,ABSVALUE:def 1;
  assume
A7: ICplusConst(s1,k1) = ICplusConst(s2,k2);
  ex j being Element of NAT st j = IC s2 & ICplusConst(s2, k2)=|.j+k2.| by
SCMPDS_2:def 18;
  then ICplusConst(s2,k2)=m+k2 by A1,A3,A5,ABSVALUE:def 1;
  hence contradiction by A2,A7,A6;
end;
