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 Th78:
  1<=i & i<=n implies (1_Rmatrix(n)).i=Base_FinSeq(n,i)
proof
  assume
A1: 1<=i & i<=n;
  then 1<=n by XXREAL_0:2;
  then
A2: 1 in Seg n;
  i in Seg n by A1;
  then [i,1] in [:Seg n, Seg n:] by A2,ZFMISC_1:87;
  then [i,1] in Indices (1.(F_Real,n)) by MATRIX_0:24;
  then consider q being FinSequence of REAL such that
A3: q = (1_Rmatrix n).i and
  (1_Rmatrix n)*(i,1) = q.1 by MATRIX_0:def 5;
  len (1_Rmatrix(n))=n by MATRIX_0:24;
  then i in dom 1_Rmatrix n by A1,FINSEQ_3:25;
  then q in rng 1_Rmatrix n by A3,FUNCT_1:def 3;
  then
A4: len q=n by MATRIX_0:def 2;
A5: for j be Nat st 1<=j & j<=n holds q.j=(Base_FinSeq(n,i)).j
  proof
A6: i in Seg n by A1;
    let j be Nat;
    assume
A7: 1<=j & j<=n;
    j in Seg n by A7;
    then [i,j] in [:Seg n, Seg n:] by A6,ZFMISC_1:87;
    then
A8: [i,j] in Indices (1.(F_Real,n)) by MATRIX_0:24;
    then
A9: ex q0 being FinSequence of REAL st q0 = (1_Rmatrix(n)).i & (1_Rmatrix(
    n))*(i,j) = q0.j by MATRIX_0:def 5;
    per cases;
    suppose
A10:  i=j;
      then (1.(F_Real,n))*(i,i) = 1_(F_Real) by A8,MATRIX_1:def 3;
      hence thesis by A1,A3,A9,A10,Th75;
    end;
    suppose
A11:  i<>j;
      then (1.(F_Real,n))*(i,j) = 0.(F_Real) by A8,MATRIX_1:def 3;
      hence thesis by A3,A7,A9,A11,Th76;
    end;
  end;
  len Base_FinSeq(n,i)=n by Th74;
  hence thesis by A3,A4,A5,FINSEQ_1:14;
end;
