 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;

theorem REALTOINT:
  for M being Matrix of D
  st for i, j being Nat st [i, j] in Indices M holds M*(i,j) in E
  holds M is Matrix of E
  proof
    let M be Matrix of D;
    assume
AS: for i, j being Nat st [i, j] in Indices M holds M*(i,j) in E;
    consider n be Nat such that
    A0: for x being object st x in rng M holds
    ex p being FinSequence of D st x = p & len p = n by MATRIX_0:9;
    per cases;
    suppose len M = 0;
      then M = {};
      then rng M c= E*;
      hence thesis by FINSEQ_1:def 4;
    end;
    suppose I0: len M <> 0;
      for x being object st x in rng M holds
      ex p being FinSequence of E st x = p & len p = n
      proof
        let x be object;
        assume S1: x in rng M;
        then consider p be FinSequence of D such that
        S3: x = p & len p = n by A0;
        X1: width M = n by S1,S3,MATRIX_0:def 3,I0;
        for z being object st z in rng p holds z in E
        proof
          let z be object;
          assume z in rng p;
          then consider j1 be object such that
          S4: j1 in dom p & z = p.j1 by FUNCT_1:def 3;
          S5: j1 in Seg n by S3,S4,FINSEQ_1:def 3;
          reconsider j1 as Nat by S4;
          consider i1 be object such that
          S6: i1 in dom M & x = M.i1 by S1,FUNCT_1:def 3;
          reconsider i1 as Nat by S6;
          S8: [i1,j1] in Indices M by S6,S5,X1,ZFMISC_1:87;
          then consider q being FinSequence of D such that
          S9: q = M.i1 & M*(i1,j1) = q.j1 by MATRIX_0:def 5;
          thus z in E by AS,S3,S4,S6,S8,S9;
        end;
        then rng p c= E;
        then reconsider p as FinSequence of E by FINSEQ_1:def 4;
        take p;
        thus x = p & len p = n by S3;
      end;
      hence M is Matrix of E by MATRIX_0:9;
    end;
  end;
