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;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;
reserve N for (Matrix of n,K),
  N1 for (Matrix of m,K);
reserve p,p1 for FinSequence of K;

theorem Th73:
  Width G1 = Len G2 implies Len (G1(#)G2) = Len G1 & Width (G1(#)
  G2) = Width G2
proof
  assume
A1: Width G1=Len G2;
  set G12=G1(#)G2;
A2: dom Len G12=dom G12 by Def3;
A3: dom G12=dom G1 by Def11;
A4: dom Width G2=dom G2 by Def4;
A5: dom Width G1=dom G1 by Def4;
A6: dom Len G1=dom G1 by Def3;
  now
    let i such that
A7: i in dom Len G12;
A8: width (G1.i) = (Width G1).i by A2,A5,A3,A7,Def4
      .= len (G2.i) by A1,A2,A5,A3,A7,Def3;
    thus (Len G12).i = len (G12.i) by A7,Def3
      .= len (G1.i*G2.i) by A2,A7,Def11
      .= len (G1.i) by A8,MATRIX_3:def 4
      .= (Len G1).i by A6,A2,A3,A7,Def3;
  end;
  hence Len G12=Len G1 by A6,A2,A3,FINSEQ_1:13;
A9: dom Width G12=dom G12 by Def4;
A10: dom Len G2=dom G2 by Def3;
  now
    let i such that
A11: i in dom Width G12;
A12: width (G1.i) = (Width G1).i by A5,A9,A3,A11,Def4
      .= len (G2.i) by A1,A5,A9,A3,A11,Def3;
    thus (Width G12).i = width (G12.i) by A11,Def4
      .= width (G1.i*G2.i) by A9,A11,Def11
      .= width (G2.i) by A12,MATRIX_3:def 4
      .= (Width G2).i by A1,A5,A10,A9,A4,A3,A11,Def4;
  end;
  hence thesis by A1,A5,A10,A9,A4,A3,FINSEQ_1:13;
end;
