reserve p1,p2,p3,p4,p5,p6,p,pc for Point of TOP-REAL 2;
reserve a,b,c,r,s for Real;

theorem Th15:
  p in LSeg(p1,p3) & p in LSeg(p2,p4) & p<>p1 & p<>p2 & p<>p3 & p
  <>p4 implies angle(p1,p,p2)=angle(p3,p,p4)
proof
  assume
A1: p in LSeg(p1,p3);
  assume
A2: p in LSeg(p2,p4);
  assume
A3: p<>p1;
  assume
A4: p<>p2;
  assume
A5: p<>p3;
  assume
A6: p<>p4;
  per cases by A1,A2,A3,A4,A5,A6,Th13;
  suppose
    angle(p1,p,p2)+angle(p2,p,p3)=PI & angle(p2,p,p3)+angle(p3,p,p4)= PI;
    hence thesis;
  end;
  suppose
A7: angle(p1,p,p2)+angle(p2,p,p3)=3*PI & angle(p2,p,p3)+angle(p3,p,p4) =PI;
    angle(p3,p,p4)>=0 by COMPLEX2:70;
    then angle(p3,p,p4)+2*PI>=0+2*PI by XREAL_1:6;
    hence thesis by A7,COMPLEX2:70;
  end;
  suppose
A8: angle(p1,p,p2)+angle(p2,p,p3)=PI & angle(p2,p,p3)+angle(p3,p,p4)= 3*PI;
    angle(p3,p,p4)<2*PI by COMPLEX2:70;
    then angle(p3,p,p4)-2*PI<2*PI-2*PI by XREAL_1:9;
    hence thesis by A8,COMPLEX2:70;
  end;
  suppose
    angle(p1,p,p2)+angle(p2,p,p3)=3*PI & angle(p2,p,p3)+angle(p3,p,p4 )=3*PI;
    hence thesis;
  end;
end;
