 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;

theorem
  for D be non empty set for M be Matrix of n,m,D holds
    M-X is Matrix of n-'card(M"X),m,D
proof
  let D be non empty set;
  let M be Matrix of n,m,D;
  A1: rng(M-X)c=rng M by FINSEQ_3:66;
  rng M c=D* by FINSEQ_1:def 4;
  then rng(M-X)c=D* by A1;
  then reconsider MX=M-X as FinSequence of D* by FINSEQ_1:def 4;
  A2: len MX=len M-card(M"X) by FINSEQ_3:59;
  then len M >= card(M"X) by XREAL_1:49;
  then
A3: len M=n & len MX=len M-' card(M"X) by A2,MATRIX_0:def 2,XREAL_1:233;
  per cases;
  suppose  len MX=0;
   then MX={};
   hence thesis by A3,MATRIX_0:13;
  end;
  suppose len MX>0;
   A4: for x be object st x in rng MX ex s be FinSequence st s=x & len s=m
   proof
    let x be object;
    consider nn be Nat such that
     A5: for x be object st x in rng M ex p be FinSequence of D st
      x=p & len p=nn by MATRIX_0:9;
    assume A6: x in rng MX;
    then ex p be FinSequence of D st x=p & len p=nn by A1,A5;
    then reconsider p=x as FinSequence of D;
    len p=m by A1,A6,MATRIX_0:def 2;
    hence thesis;
   end;
   then reconsider MX as Matrix of D by MATRIX_0:def 1;
   now let p be FinSequence of D;
    assume p in rng MX;
    then ex s be FinSequence st s=p & len s=m by A4;
    hence len p=m;
   end;
   hence thesis by A3,MATRIX_0:def 2;
  end;
end;
