reserve x,y for object,
  N for Element of NAT,
  c,i,j,k,m,n for Nat,
  D for non empty set,
  s for Element of 2Set Seg (n+2),
  p for Element of Permutations(n) ,
  p1, q1 for Element of Permutations(n+1),
  p2 for Element of Permutations(n +2),
  K for Field,
  a for Element of K,
  f for FinSequence of K,
  A for (Matrix of K),
  AD for Matrix of n,m,D,
  pD for FinSequence of D,
  M for Matrix of n,K;

theorem
  RCol(AD,c,Col(AD,c)) = AD
proof
  set C=Col(AD,c);
  set R=RCol(AD,c,C);
  now
    reconsider c as Element of NAT by ORDINAL1:def 12;
    let i,j be Nat such that
A1: [i,j] in Indices AD;
A2: len C=len AD by MATRIX_0:def 8;
    reconsider I=i,J=j as Element of NAT by ORDINAL1:def 12;
A3: i in dom AD by A1,ZFMISC_1:87;
    now
      per cases;
      suppose
A4:     c = j;
        hence R*(i,j) = C.I by A1,A2,Def2
          .= AD*(i,j) by A3,A4,MATRIX_0:def 8;
      end;
      suppose
        c <> J;
        hence R*(i,j)=AD*(i,j) by A1,A2,Def2;
      end;
    end;
    hence R*(i,j)=AD*(i,j);
  end;
  hence thesis by MATRIX_0:27;
end;
