reserve V for RealLinearSpace;
reserve p,q,r,u,v,w,y,u1,v1,w1 for Element of V;
reserve a,b,c,d,a1,b1,c1,a2,b2,c2,a3,b3,e,f for Real;

theorem
  not are_Prop p,q & a1*p + b1*q = a2*p + b2*q & p is not zero & q is
  not zero implies a1 = a2 & b1 = b2
proof
  assume that
A1: not are_Prop p,q and
A2: a1*p + b1*q = a2*p + b2*q and
A3: p is not zero and
A4: q is not zero;
  a2*p + b2*q + -b1*q = a1*p + (b1*q + -b1*q) by A2,RLVECT_1:def 3
    .= a1*p + 0.V by RLVECT_1:5
    .= a1*p;
  then a1*p = (b2*q + -b1*q) + a2*p by RLVECT_1:def 3
    .= (b2*q - b1*q) + a2*p by RLVECT_1:def 11
    .= (b2-b1)*q + a2*p by RLVECT_1:35;
  then a1*p + -a2*p = (b2-b1)*q + (a2*p + -a2*p) by RLVECT_1:def 3
    .= (b2-b1)*q + 0.V by RLVECT_1:5
    .= (b2-b1)*q;
  then
A5: (b2-b1 )*q = a1*p - a2*p by RLVECT_1:def 11
    .= (a1-a2)*p by RLVECT_1:35;
A6: q<>0.V by A4;
A7: now
    assume
A8: a1-a2=0;
    then (b2-b1)*q = 0.V by A5,RLVECT_1:10;
    then b2-b1=0 by A6,RLVECT_1:11;
    hence thesis by A8;
  end;
A9: p<>0.V by A3;
  now
    assume
A10: b2-b1=0;
    then (a1-a2)*p = 0.V by A5,RLVECT_1:10;
    then a1-a2=0 by A9,RLVECT_1:11;
    hence thesis by A10;
  end;
  hence thesis by A1,A5,A7;
end;
