reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;

theorem Th42:
  [[u,v],[u1,v1]] in DTrapezium(V,w,y) iff u,v,u1,v1 are_DTr_wrt w ,y
proof
  now
    assume [[u,v],[u1,v1]] in DTrapezium(V,w,y);
    then consider u9,v9,u19,v19 being VECTOR of V such that
A1: [u,v]=[u9,v9] and
A2: [u1,v1]=[u19,v19] and
A3: u9,v9,u19,v19 are_DTr_wrt w,y by Def7;
A4: u1=u19 by A2,XTUPLE_0:1;
    u=u9 & v=v9 by A1,XTUPLE_0:1;
    hence u,v,u1,v1 are_DTr_wrt w,y by A2,A3,A4,XTUPLE_0:1;
  end;
  hence thesis by Def7;
end;
