reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th68:
  for A being Matrix of K holds (1.(K,len A))*A=A
proof
  let A be Matrix of K;
  set n=len A;
  set B=1.(K,n);
A1: len B=n by MATRIX_0:24;
A2: width B=n by MATRIX_0:24;
  then
A3: len (B*A)=len B by MATRIX_3:def 4;
A4: now
A5: dom A = Seg len A by FINSEQ_1:def 3;
    let i,j be Nat;
    assume
A6: [i,j] in Indices (B*A);
A7: dom (B*A) = Seg len (B*A) by FINSEQ_1:def 3;
    then
A8: i in Seg width B by A1,A2,A3,A6,ZFMISC_1:87;
    then i in Seg len(Line(B,i)) by MATRIX_0:def 7;
    then
A9: i in dom (Line(B,i)) by FINSEQ_1:def 3;
A10: dom B = Seg len B by FINSEQ_1:def 3;
    then
A11: i in dom B by A3,A6,A7,ZFMISC_1:87;
    then [i,i] in Indices B by A8,ZFMISC_1:87;
    then
A12: (Line(B,i)).i=1_K by MATRIX_3:15;
    i in Seg len (Col(A,j)) by A2,A8,MATRIX_0:def 8;
    then
A13: i in dom Col(A,j) by FINSEQ_1:def 3;
A14: now
      let k be Nat;
      assume that
A15:  k in dom (Line (B,i)) and
A16:  k<>i;
      k in Seg len (Line (B,i)) by A15,FINSEQ_1:def 3;
      then k in Seg width B by MATRIX_0:def 7;
      then [i,k] in Indices B by A11,ZFMISC_1:87;
      hence (Line(B,i)).k=0.K by A16,MATRIX_3:15;
    end;
    thus (B*A)*(i,j)= Line(B,i) "*" Col(A,j) by A2,A6,MATRIX_3:def 4
      .=Sum(mlt(Line(B,i),Col(A,j))) by FVSUM_1:def 9
      .=Col(A,j).i by A9,A13,A14,A12,MATRIX_3:17
      .=A*(i,j) by A1,A5,A10,A11,MATRIX_0:def 8;
  end;
  width (B*A)=width A by A2,MATRIX_3:def 4;
  hence thesis by A1,A3,A4,MATRIX_0:21;
end;
