reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;
reserve R for Ring;

theorem Th79:
  0 < m implies
  for M being Matrix of m,1,F_Real holds M is FinSequence of 1-tuples_on REAL
  proof
    assume
A0: 0 < m;
    let M be Matrix of m,1,F_Real;
A1: len M = m by A0,MATRIX_0:23;
    width M = 1 by A0,MATRIX_0:23;
    then consider s be FinSequence such that
A2: s in rng M and
A3: len s = 1 by A0,A1,MATRIX_0:def 3;
    consider n be Nat such that
A4: for x be object st x in rng M
    ex s be FinSequence st s = x & len s = n by MATRIX_0:def 1;
    consider s1 be FinSequence such that
A5: s1 = s and
A6: len s1 = n by A4,A2;
    rng M c= 1-tuples_on REAL
    proof
      let x be object;
      assume
A7:   x in rng M;
      then consider s being FinSequence such that
A8:   s = x and
A9:   len s = n by A4;
      consider n0 be Nat such that
A10:  for x be object st x in rng M ex p be FinSequence of F_Real st
        x = p & len p = n0 by MATRIX_0:9;
      consider p be FinSequence of F_Real such that
A11:  x = p and
      len p = n0 by A10,A7;
      rng p c= REAL;
      hence thesis by A11,A8,A9,A5,A6,A3,FINSEQ_2:132;
    end;
    hence thesis by FINSEQ_1:def 4;
  end;
