reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;

theorem Th21:
  Width (F1|n) =(Width F1) |n
proof
A1: len Width F1=len F1 by CARD_1:def 7;
  per cases;
  suppose
A2: n>=len F1;
    then F1|n=F1 by FINSEQ_1:58;
    hence thesis by A1,A2,FINSEQ_1:58;
  end;
  suppose
A3: n<len F1;
    F1=(F1|n)^(F1/^n) by RFINSEQ:8;
    then
A4: Width F1=(Width (F1|n))^Width (F1/^n) by Th18;
    len (F1|n) =n by A3,FINSEQ_1:59;
    then len Width (F1|n)=n by CARD_1:def 7;
    hence thesis by A4,FINSEQ_5:23;
  end;
end;
