reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  W for Element of V;
reserve KL1,KL2,KL3 for Linear_Combination of V,
  X for Subset of V;
reserve s for FinSequence,
  V1,V2,V3 for finite-dimensional VectSp of K,
  f,f1,f2 for Function of V1,V2,
  g for Function of V2,V3,
  b1 for OrdBasis of V1,
  b2 for OrdBasis of V2,
  b3 for OrdBasis of V3,
  v1,v2 for Vector of V2,
  v,w for Element of V1;
reserve p2,F for FinSequence of V1,
  p1,d for FinSequence of K,
  KL for Linear_Combination of V1;

theorem Th15:
  for x be Element of D holds <*<*x*>*> = <*<*x*>*>@
proof
  let x be Element of D;
  set P = <*<*x*>*>, R = (<*<*x*>*>@);
A1: len P = 1 by FINSEQ_1:40;
  then
A2: width P = len <*x*> by MATRIX_0:20
    .= 1 by FINSEQ_1:40;
  then
A3: len R = 1 by MATRIX_0:54;
A4: now
    let i,j;
    assume
A5: [i,j] in Indices P;
    then
A6: [i,j] in [:dom P,Seg 1:] by A2,MATRIX_0:def 4;
    then i in dom P by ZFMISC_1:87;
    then i in Seg 1 by A1,FINSEQ_1:def 3;
    then
A7: i = 1 by FINSEQ_1:2,TARSKI:def 1;
    j in Seg 1 by A6,ZFMISC_1:87;
    then j = 1 by FINSEQ_1:2,TARSKI:def 1;
    hence P*(i,j) = R*(i,j) by A5,A7,MATRIX_0:def 6;
  end;
  width R = 1 by A1,A2,MATRIX_0:54;
  hence thesis by A1,A2,A3,A4,MATRIX_0:21;
end;
