reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;
reserve rseq1,rseq2 for convergent Real_Sequence;
reserve n,m,N,M for Nat;
reserve e,r for Real;
reserve Pseq for P-convergent Function of [:NAT,NAT:],REAL;

theorem
for Rseq be Function of [:NAT,NAT:],REAL, r be Real st
  Rseq is P-convergent holds
 r(#)Rseq is P-convergent & P-lim (r(#)Rseq) = r * P-lim Rseq
proof
   let Rseq be Function of [:NAT,NAT:],REAL;
   let r be Real;
   assume a1: Rseq is P-convergent;
a4:now assume a2: r=0;
a3: now let n,m;
     (r(#)Rseq).(n,m) = r * Rseq.(n,m) by VALUED_1:6;
     hence (r(#)Rseq).(n,m) = 0 by a2;
    end;
    hence r(#)Rseq is P-convergent by lm55a;
    thus P-lim (r(#)Rseq) = 0 by a3,lm55a;
   end;
   now assume r <> 0; then
a5: |.r.| > 0 by COMPLEX1:47;
a7: now let e be Real;
     assume 0 < e; then
     consider N such that
a6:   for n,m st n>=N & m>=N holds |. Rseq.(n,m) - P-lim Rseq.| < e/ |.r.|
        by a1,a5,def6;
     take N;
     hereby let n,m;
      assume n>=N & m>=N; then
      |. Rseq.(n,m) - P-lim Rseq.| < e/ |.r.| by a6; then
      |.r.| * |.Rseq.(n,m) - P-lim Rseq.| < e/ |.r.| * |.r.|
        by a5,XREAL_1:68; then
      |.r.| * |. Rseq.(n,m) - P-lim Rseq.| < |.r.| / (|.r.| / e)
        by XCMPLX_1:79; then
      |.r.| * |. Rseq.(n,m) - P-lim Rseq.| < e by a5,XCMPLX_1:52; then
      |.r*(Rseq.(n,m) - P-lim Rseq).| < e by COMPLEX1:65; then
      |.r*Rseq.(n,m) - r * P-lim Rseq.| < e;
      hence |.(r(#)Rseq).(n,m) - r * P-lim Rseq.| < e by VALUED_1:6;
     end;
    end;
    hence r(#)Rseq is P-convergent;
    hence P-lim (r(#)Rseq) = r * P-lim Rseq by a7,def6;
   end;
   hence thesis by a4;
end;
