reserve x,y,z,x1,x2,y1,y2,z1,z2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Ring;

theorem
  for K being Ring
  for a,b being Element of K holds <*<*a*>*> * <*<*b*>*> =<*<*a*b*>*>
proof
  let K be Ring;
  let a,b be Element of K;
  reconsider A=<*<*a*>*> as Matrix of 1,K;
  reconsider B=<*<*b*>*> as Matrix of 1,K;
  reconsider C=A*B as Matrix of K;
A1: len Line(A,1)=width A by MATRIX_0:def 7
    .=1 by MATRIX_0:24;
A2: width A=1 by MATRIX_0:24;
  then 1 in Seg width A by FINSEQ_1:2,TARSKI:def 1;
  then Line(A,1).1=<*<*a*>*>*(1,1) by MATRIX_0:def 7
    .=a by MATRIX_0:49;
  then
A3: Line(<*<*a*>*>,1)=<*a*> by A1,FINSEQ_1:40;
A4: len B=1 by MATRIX_0:24;
  then 1 in Seg len B by FINSEQ_1:2,TARSKI:def 1;
  then 1 in dom B by FINSEQ_1:def 3;
  then
A5: Col(B,1).1=<*<*b*>*>*(1,1) by MATRIX_0:def 8
    .=b by MATRIX_0:49;
  len A =1 by MATRIX_0:24;
  then
A6: len C=1 by A2,A4,Def4;
  width B=1 by MATRIX_0:24;
  then
A7: width C=1 by A2,A4,Def4;
  then reconsider C as Matrix of 1,K by A6,MATRIX_0:51;
  Seg len C = dom C by FINSEQ_1:def 3;
  then
A8: Indices C=[:Seg 1,Seg 1:] by A6,A7;
  len Col(B,1)=len B by MATRIX_0:def 8
    .=1 by MATRIX_0:24;
  then
A9: Col(<*<*b*>*>,1)=<*b*> by A5,FINSEQ_1:40;
  now
    let i,j;
    assume
A10: [i,j] in Indices C;
    then j in Seg 1 by A8,ZFMISC_1:87;
    then
A11: j=1 by FINSEQ_1:2,TARSKI:def 1;
    i in Seg 1 by A8,A10,ZFMISC_1:87;
    then
A12: i=1 by FINSEQ_1:2,TARSKI:def 1;
    hence C*(i,j)=<*a*> "*" <*b*> by A2,A4,A3,A9,A10,A11,Def4
      .= a * b by FVSUM_1:88
      .=<*<*a*b*>*>*(i,j) by A12,A11,MATRIX_0:49;
  end;
  hence thesis by MATRIX_0:27;
end;
