reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

theorem Th57:
  for u,v being non zero Element of TOP-REAL 3
  for a,b being Real st (a <> 0 or b <> 0) & a * u + b * v = 0.TOP-REAL 3
  holds are_Prop u,v
  proof
    let u,v be non zero Element of TOP-REAL 3;
    let a,b be Real;
    assume that
A1: a <> 0 or b <> 0 and
A2: a * u + b * v = 0.TOP-REAL 3;
    reconsider n = 3 as Nat;
    reconsider au = a * u, bv = b * v as Element of TOP-REAL 3;
    consider c be Real such that
A3: c <> 0 and
A4: au = c * bv by A2,ANPROJ_1:1,13;
A5: a <> 0 & b <> 0
    proof
      assume a = 0 or b = 0;
      then per cases;
      suppose
A6:     a = 0;
        u in TOP-REAL 3;
        then u in REAL 3 by EUCLID:22; then
A7:     0 * u = 0*n by EUCLID_4:3;
        bv in TOP-REAL 3;
        then bv in REAL 3 by EUCLID:22;
        then 0*n + bv = bv by EUCLID_4:1;
        hence contradiction by A1,A6,A2,A7,Th56;
      end;
      suppose
A8:     b = 0;
        v in TOP-REAL 3;
        then v in REAL 3 by EUCLID:22; then
A9:     0 * v = 0*n by EUCLID_4:3;
        au in TOP-REAL 3;
        then au in REAL 3 by EUCLID:22;
        then au + 0*n = au by EUCLID_4:1;
        hence contradiction by A1,A8,A2,A9,Th56;
      end;
    end;
    u = 1 * u by RVSUM_1:52
     .= (1/a * a) * u by A5,XCMPLX_1:106
     .= (1/a) * (c * (b * v)) by A4,RVSUM_1:49
     .= ((1/a * c) * (b * v)) by RVSUM_1:49
     .= (1/a * c * b) * v by RVSUM_1:49;
    hence thesis by A3,A5,ANPROJ_1:1;
  end;
