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 Th51:
  for n being Element of NAT, p1,p2,p3,p0 being Point of TOP-REAL
  n st p2-p1,p3-p1 are_lindependent2 & p0 in plane(p1,p2,p3)
 ex a1,a2,a3 being Real
   st p0=a1*p1+a2*p2+a3*p3 & a1+a2+a3=1 &
for b1,b2,b3 being Real st p0
  =b1*p1+b2*p2+b3*p3 & b1+b2+b3=1 holds b1=a1 & b2=a2 & b3=a3
proof
  let n be Element of NAT,p1,p2,p3,p0 be Point of TOP-REAL n;
  assume that
A1: p2-p1,p3-p1 are_lindependent2 and
A2: p0 in plane(p1,p2,p3);
  set q2=p2-p1,q3=p3-p1;
  consider a01,a02,a03 being Real such that
A3: a01+a02+a03=1 and
A4: p0=a01*p1+a02*p2+a03*p3 by A2,Th48;
  for b1,b2,b3 being Real
st p0=b1*p1+b2*p2+b3*p3 & b1+b2+b3=1 holds b1=
  a01 & b2=a02 & b3=a03
  proof
A5: p0+-p1=a01*p1+a02*p2+a03*p3+-(a01+a02+a03)*p1 by A3,A4,RLVECT_1:def 8
      .=a01*p1+a02*p2+a03*p3+-((a01+a02)*p1+a03*p1) by RLVECT_1:def 6
      .=a01*p1+a02*p2+a03*p3+-(a01*p1+a02*p1+a03*p1) by RLVECT_1:def 6
      .=a01*p1+a02*p2+a03*p3+(-(a01*p1+a02*p1)-a03*p1) by RLVECT_1:30
      .=a01*p1+a02*p2+a03*p3+(-a01*p1-a02*p1-a03*p1) by RLVECT_1:30
      .=a01*p1+(a02*p2+a03*p3)+(-a01*p1+-a02*p1+-a03*p1) by RLVECT_1:def 3
      .=a01*p1+(a02*p2+a03*p3)+(-a01*p1+(-a02*p1+-a03*p1)) by RLVECT_1:def 3
      .=(a02*p2+a03*p3)+a01*p1-a01*p1+(-a02*p1+-a03*p1) by RLVECT_1:def 3
      .=(a02*p2+a03*p3)+(-a02*p1+-a03*p1) by RLVECT_4:1
      .=a02*p2+a03*p3+-a02*p1+-a03*p1 by RLVECT_1:def 3
      .=a02*p2+-a02*p1+a03*p3+-a03*p1 by RLVECT_1:def 3
      .=a02*p2+a02*(-p1)+a03*p3+-a03*p1 by RLVECT_1:25
      .=a02*(p2+(-p1))+a03*p3+-a03*p1 by RLVECT_1:def 5
      .=a02*(p2+(-p1))+(a03*p3+-a03*p1) by RLVECT_1:def 3
      .=a02*(p2+(-p1))+(a03*p3+a03*(-p1)) by RLVECT_1:25
      .=a02*q2+a03*q3 by RLVECT_1:def 5;
    let b1,b2,b3 be Real;
    assume that
A6: p0=b1*p1+b2*p2+b3*p3 and
A7: b1+b2+b3=1;
    p0+-p1=b1*p1+b2*p2+b3*p3+-(b1+b2+b3)*p1 by A6,A7,RLVECT_1:def 8
      .=b1*p1+b2*p2+b3*p3+-((b1+b2)*p1+b3*p1) by RLVECT_1:def 6
      .=b1*p1+b2*p2+b3*p3+-(b1*p1+b2*p1+b3*p1) by RLVECT_1:def 6
      .=b1*p1+b2*p2+b3*p3+(-(b1*p1+b2*p1)-b3*p1) by RLVECT_1:30
      .=b1*p1+b2*p2+b3*p3+(-b1*p1-b2*p1-b3*p1) by RLVECT_1:30
      .=b1*p1+(b2*p2+b3*p3)+(-b1*p1+-b2*p1+-b3*p1) by RLVECT_1:def 3
      .=b1*p1+(b2*p2+b3*p3)+(-b1*p1+(-b2*p1+-b3*p1)) by RLVECT_1:def 3
      .=(b2*p2+b3*p3)+b1*p1-b1*p1+(-b2*p1+-b3*p1) by RLVECT_1:def 3
      .=(b2*p2+b3*p3)+(-b2*p1+-b3*p1) by RLVECT_4:1
      .=b2*p2+b3*p3+-b2*p1+-b3*p1 by RLVECT_1:def 3
      .=b2*p2+-b2*p1+b3*p3+-b3*p1 by RLVECT_1:def 3
      .=b2*p2+b2*(-p1)+b3*p3+-b3*p1 by RLVECT_1:25
      .=b2*(p2+(-p1))+b3*p3+-b3*p1 by RLVECT_1:def 5
      .=b2*(p2+(-p1))+(b3*p3+-b3*p1) by RLVECT_1:def 3
      .=b2*(p2+(-p1))+(b3*p3+b3*(-p1)) by RLVECT_1:25
      .=b2*q2+b3*q3 by RLVECT_1:def 5;
    then b2*q2+b3*q3+-(a02*q2+a03*q3)=0.TOP-REAL n by A5,RLVECT_1:5;
    then b2*q2+b3*q3+(-a02*q2-a03*q3)=0.TOP-REAL n by RLVECT_1:30;
    then b2*q2+b3*q3+-a02*q2+-a03*q3=0.TOP-REAL n by RLVECT_1:def 3;
    then b2*q2+-a02*q2+b3*q3+-a03*q3=0.TOP-REAL n by RLVECT_1:def 3;
    then b2*q2+(-a02)*q2+b3*q3+-a03*q3=0.TOP-REAL n by RLVECT_1:79;
    then (b2+-a02)*q2+b3*q3+-a03*q3=0.TOP-REAL n by RLVECT_1:def 6;
    then (b2+-a02)*q2+(b3*q3+-a03*q3)=0.TOP-REAL n by RLVECT_1:def 3;
    then (b2+-a02)*q2+(b3*q3+(-a03)*q3)=0.TOP-REAL n by RLVECT_1:79;
    then (b2+-a02)*q2+(b3+-a03)*q3=0.TOP-REAL n by RLVECT_1:def 6;
    then b2-a02+a02=0+a02 & b3+-a03=0 by A1;
    hence thesis by A3,A7;
  end;
  hence thesis by A3,A4;
end;
