reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;
reserve fr,f for FinSequence of INT;
reserve b,m for Nat;
reserve b for Integer;
reserve m for Integer;
reserve fp for FinSequence of NAT;

theorem
  p > 2 & d gcd p = 1 & e gcd p = 1 implies Lege((d^2)*e,p) = Lege(e,p)
proof
  assume that
A1: p > 2 and
A2: d gcd p = 1 and
A3: e gcd p = 1;
  reconsider d2=d^2, e as Element of NAT by ORDINAL1:def 12;
  set fp = <*d2,e*>;
  reconsider fp as FinSequence of NAT by FINSEQ_2:13;
  not p divides d by A2,Lm3; then
  d mod p <> 0 by INT_1:62; then
A4: Lege(d^2,p) = 1 by Th26;
  reconsider p as prime Element of NAT by ORDINAL1:def 12;
  for k st k in dom fp holds fp.k gcd p = 1
  proof
    let k;
    assume k in dom fp;
    then k in Seg(len fp) by FINSEQ_1:def 3;
    then
A5: k in Seg 2 by FINSEQ_1:44;
    per cases by A5,FINSEQ_1:2,TARSKI:def 2;
    suppose
      k = 1;
      then fp.k = d^2;
      hence thesis by A2,WSIERP_1:7;
    end;
    suppose
      k = 2;
      hence thesis by A3;
    end;
  end;
  then consider fr be FinSequence of INT such that
A6: len fr = len fp and
A7: for k be Nat st k in dom fr holds fr.k = Lege (fp.k,p) and
A8: Lege (Product fp,p) = Product fr by A1,Th34;
A9: len fr = 2 by A6,FINSEQ_1:44;
  then 2 in dom fr by FINSEQ_3:25;
  then fr.2 = Lege(fp.2,p) by A7;
  then
A10: fr.2 = Lege(e,p);
  fr.1 = Lege(fp.1,p) by A7,A9,FINSEQ_3:25;
  then fr.1 = Lege(d^2,p);
  then fr = <*1,Lege(e,p)*> by A4,A9,A10,FINSEQ_1:44;
  then Product fr = 1 * Lege(e,p) by RVSUM_1:99;
  hence thesis by A8,RVSUM_1:99;
end;
