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;
reserve x,y,z for object;

theorem Th20:
  [u,v] in Proportionality_as_EqRel_of V iff u is not zero & v is
  not zero & are_Prop u,v
proof
A1: now
    assume
A2: [u,v] in Proportionality_as_EqRel_of V;
    then u in NonZero V & v in NonZero V by Def3;
    hence u is not zero & v is not zero by STRUCT_0:1;
    ex u1,v1 st u=u1 & v=v1 & are_Prop u1,v1 by A2,Def3;
    hence are_Prop u,v;
  end;
  now
    assume that
A3: u is not zero & v is not zero and
A4: are_Prop u,v;
    u in NonZero V & v in NonZero V by A3,STRUCT_0:1;
    hence [u,v] in Proportionality_as_EqRel_of V by A4,Def3;
  end;
  hence thesis by A1;
end;
