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 Th18:
  Width (F1 ^ F2)= Width F1 ^ Width F2
proof
  set F12=F1^F2;
A1: len F12=len F1+len F2 by FINSEQ_1:22;
  len F2=len Width F2 by CARD_1:def 7;
  then
A2: dom (Width F2)=dom F2 by FINSEQ_3:29;
A3: len (Width F1^Width F2)=len F1+len F2 by CARD_1:def 7;
A4: len Width F12=len F12 by CARD_1:def 7;
  then
A5: dom (Width F1 ^ Width F2) =dom Width F12 by A1,A3,FINSEQ_3:29;
A6: len F1=len Width F1 by CARD_1:def 7;
  then
A7: dom (Width F1)=dom F1 by FINSEQ_3:29;
  now
    let k such that
A8: 1<=k and
A9: k<=len F1+len F2;
A10: k in dom (Width F1 ^ Width F2) by A3,A8,A9,FINSEQ_3:25;
    now
      per cases by A10,FINSEQ_1:25;
      suppose
A11:    k in dom (Width F1);
        hence (Width F1 ^ Width F2).k = (Width F1).k by FINSEQ_1:def 7
          .= width (F1.k) by A11,Def4
          .= width(F12.k) by A7,A11,FINSEQ_1:def 7
          .= (Width F12).k by A5,A10,Def4;
      end;
      suppose
        ex n st n in dom (Width F2) & k=len (Width F1)+n;
        then consider n such that
A12:    n in dom (Width F2) and
A13:    k=len F1+n by A6;
        thus (Width F1 ^ Width F2).k = (Width F2).n by A6,A12,A13,
FINSEQ_1:def 7
          .= width (F2.n) by A12,Def4
          .= width(F12.k) by A2,A12,A13,FINSEQ_1:def 7
          .= (Width F12).k by A5,A10,Def4;
      end;
    end;
    hence (Width F12).k=(Width F1 ^ Width F2).k;
  end;
  hence thesis by A4,A1,A3;
end;
