reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;
reserve fp,fp1 for FinSequence of NAT,

  b,c,d, n for Element of NAT,
  a for Nat;
reserve i,m,m1,m2,m3,r,s,a,b,c,c1,c2,x,y for Integer;
reserve

  a,b,c,m for Element of NAT;
reserve a for Integer;

theorem
  for X being finite Subset of INT st X is_CRS_of m holds (a ++ X) is_CRS_of m
proof
  let X be finite Subset of INT;
  assume
A1: X is_CRS_of m;
  then card X = m by Th49;
  then
A2: card (a ++ X) = m by Th2;
A3: for i,j being Integer st i in a++X & j in a++X & i<>j holds not [i,j] in
  Cong(m)
  proof
    let i,j be Integer;
    assume that
A4: i in a++X and
A5: j in a++X and
A6: i<>j;
    consider u being Complex such that
A7: i=a+u and
A8: u in X by A4,MEMBER_1:143;
    consider w being Complex such that
A9: j=a+w and
A10: w in X by A5,MEMBER_1:143;
    reconsider u9=u, w9=w as Integer by A8,A10;
    assume [i,j] in Cong(m);
    then i,j are_congruent_mod m by Def1;
    then m divides ((a+u9)-(a+w9)) by A7,A9;
    then m divides (u9-w9);
    then
A11: u9,w9 are_congruent_mod m;
    not [u9,w9] in Cong(m) by A1,A6,A8,A7,A10,A9,Th49;
    hence contradiction by A11,Def1;
  end;
  a ++ X is Subset of INT by Lm1;
  hence thesis by A2,A3,Th52;
end;
