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

theorem Th86:
  for x being FinSequence of REAL st len x=n & n>0 holds ( 1_Rmatrix n)*x=x
proof
  let x be FinSequence of REAL;
  assume that
A1: len x=n and
A2: n>0;
A3: len (ColVec2Mx x)=len x by A1,A2,MATRIXR1:def 9;
A4: len Col((ColVec2Mx x),1) = len (ColVec2Mx x) by MATRIX_0:def 8;
A5: for k being Nat st 1 <=k & k <= len (Col((ColVec2Mx x),1)) holds (Col((
  ColVec2Mx x),1)).k=x.k
  proof
    let k be Nat;
    assume
A6: 1 <=k & k <= len (Col((ColVec2Mx x),1));
A7: k in Seg len ColVec2Mx x by A4,A6;
    then
A8: k in dom (ColVec2Mx x) by FINSEQ_1:def 3;
    then
A9: Col((ColVec2Mx x),1).k = (ColVec2Mx x)*(k,1) by MATRIX_0:def 8;
    1 in Seg 1 & Indices (ColVec2Mx x)=[:dom (ColVec2Mx x),Seg 1:] by A1,A2,
MATRIXR1:def 9;
    then [k,1] in Indices (ColVec2Mx x) by A8,ZFMISC_1:87;
    then consider p being FinSequence of REAL such that
A10: p=(ColVec2Mx x).k and
A11: (ColVec2Mx x)*(k,1)=p.1 by MATRIX_0:def 5;
    k in dom x by A3,A7,FINSEQ_1:def 3;
    then p=<* x.k *> by A1,A2,A10,MATRIXR1:def 9;
    hence thesis by A9,A11;
  end;
  (1_Rmatrix n)*x =Col(MXF2MXR (MXR2MXF (ColVec2Mx x)),1) by A1,A3,Th68
    .=x by A3,A4,A5;
  hence thesis;
end;
