reserve l, m, n for Nat,
  i,j,k for Instruction of SCMPDS,
  I,J,K for Program of SCMPDS,
  p,q,r for PartState of SCMPDS;
reserve a,b,c for Int_position,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer;
reserve x for set;
reserve l,l1,loc for Nat;
reserve l1,l2 for Nat,
  i1,i2 for Instruction of SCMPDS;
reserve l for Nat;

theorem Th25:
  for s1,s2 being State of SCMPDS, n,m being Nat,k1 be
Integer st IC s1= m & m+k1>=0 & IC s1 + n = IC s2
 holds ICplusConst(s1,k1) +n = ICplusConst(s2,k1)
proof
  let s1,s2 be State of SCMPDS, n,m be Nat,k1 be Integer;
  assume that
A1: IC s1= m and
A2: m+k1>=0 and
A3: IC s1 + n = IC s2;
  reconsider nk = ICplusConst(s1,k1) as Element of NAT;
  reconsider mk=m+k1 as Element of NAT by A2,INT_1:3;
  ex n1 be Element of NAT st n1 = IC s1 & ICplusConst(s1, k1) = |.n1+k1.|
  by SCMPDS_2:def 18;
  then
  (ex n2 be Element of NAT st n2 = IC s2 & ICplusConst(s2, k1) = |.n2+k1.|
   )& nk=mk by A1,ABSVALUE:def 1,SCMPDS_2:def 18;
  hence thesis by A1,A3,ABSVALUE:def 1;
end;
