reserve z,z1,z2 for Complex;
reserve r,x1,x2 for Real;
reserve p0,p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem
  for n being Element of NAT,p1,p2,p3 being Point of TOP-REAL n holds
  Triangle(p1,p2,p3) c= closed_inside_of_triangle(p1,p2,p3)
proof
  let n be Element of NAT,p1,p2,p3 be Point of TOP-REAL n;
  LSeg(p1,p2) \/ LSeg(p2,p3) \/ LSeg(p3,p1)
    c= closed_inside_of_triangle(p1,p2,p3)
  proof
    let x be object;
    assume
A1: x in LSeg(p1,p2) \/ LSeg(p2,p3) \/ LSeg(p3,p1);
    then reconsider p0=x as Point of TOP-REAL n;
A2: x in LSeg(p1,p2) \/ LSeg(p2,p3) or x in LSeg(p3,p1) by A1,XBOOLE_0:def 3;
    now
      per cases by A2,XBOOLE_0:def 3;
      case
        x in LSeg(p1,p2);
        then consider lambda being Real such that
A3:     x=(1-lambda)*p1 + lambda*p2 and
A4:     0 <= lambda and
A5:     lambda <= 1;
A6:     p0=(1-lambda)*p1 + lambda*p2+0.TOP-REAL n by A3,RLVECT_1:4
          .=(1-lambda)*p1 + lambda*p2+(0)*p3 by RLVECT_1:10;
A7:     (1-lambda)+lambda+0=1;
        1-lambda>=0 by A5,XREAL_1:48;
        hence
        ex a1,a2,a3 being Real
        st 0<=a1 & 0<=a2 & 0<=a3 & a1+a2+a3=1 & p0
        =a1*p1+a2*p2+a3*p3 by A4,A7,A6;
      end;
      case
        x in LSeg(p2,p3);
        then consider lambda being Real such that
A8:     x=(1-lambda)*p2 + lambda*p3 and
A9:     0 <= lambda and
A10:    lambda <= 1;
A11:    p0=0.TOP-REAL n +(1-lambda)*p2 + lambda*p3 by A8,RLVECT_1:4
          .=(0)*p1+(1-lambda)*p2 + lambda*p3 by RLVECT_1:10;
A12:    0+(1-lambda)+lambda=1;
        1-lambda>=0 by A10,XREAL_1:48;
        hence
        ex a1,a2,a3 being Real
       st 0<=a1 & 0<=a2 & 0<=a3 & a1+a2+a3=1 & p0
        =a1*p1+a2*p2+a3*p3 by A9,A12,A11;
      end;
      case
        x in LSeg(p3,p1);
        then consider lambda being Real such that
A13:    x=(1-lambda)*p3 + lambda*p1 and
A14:    0 <= lambda and
A15:    lambda <= 1;
A16:    p0=lambda*p1+0.TOP-REAL n+ (1-lambda)*p3 by A13,RLVECT_1:4
          .=lambda*p1+(0)*p2+ (1-lambda)*p3 by RLVECT_1:10;
A17:    lambda+0+(1-lambda)=1;
        1-lambda>=0 by A15,XREAL_1:48;
        hence
        ex a1,a2,a3 being Real
st 0<=a1 & 0<=a2 & 0<=a3 & a1+a2+a3=1 & p0
        =a1*p1+a2*p2+a3*p3 by A14,A17,A16;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
