reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;
reserve L,L0,L1,L2 for Element of line_of_REAL n;
reserve P,P0,P1,P2 for Element of plane_of_REAL n;

theorem Th92:
  x1 in P & x2 in P & x3 in P & x2 - x1, x3 - x1 are_lindependent2
  implies P = plane(x1,x2,x3)
proof
  assume that
A1: x1 in P and
A2: x2 in P and
A3: x3 in P and
A4: x2 - x1, x3 - x1 are_lindependent2;
  P in plane_of_REAL n;
  then
  ex P9 being Subset of REAL n st P= P9 & ex y1,y2,y3 being Element of REAL
  n st P9 = plane(y1,y2,y3);
  then consider y1,y2,y3 being Element of REAL n such that
A5: P = plane(y1,y2,y3);
  ex x9 be Element of REAL n st x2 = x9 &
ex a19,a29,a39 being Real st a19
  +a29+a39=1 & x9 = a19*y1+a29*y2+a39* y3 by A2,A5;
  then consider a21,a22,a23 being Real such that
A6: a21+a22+a23=1 & x2 = a21*y1+a22*y2+a23*y3;
  ex x9 be Element of REAL n st x1 = x9 &
   ex a19,a29,a39 being Real st a19+
  a29+a39=1 & x9 = a19*y1+a29*y2+a39* y3 by A1,A5;
  then consider a11,a12,a13 being Real such that
A7: a11+a12+a13=1 & x1 = a11*y1+a12*y2+a13*y3;
  ex x9 be Element of REAL n st x3 = x9 &
  ex a19,a29,a39 being Real st a19
  +a29+a39=1 & x9 = a19*y1+a29*y2+a39 *y3 by A3,A5;
  then consider a31,a32,a33 being Real such that
A8: a31+a32+a33=1 & x3 = a31*y1+a32*y2+a33*y3;
  x3 = y1 + a32*(y2 - y1) + a33*(y3 - y1) by A8,Th27;
  then
A9: x3 - x1 = y1 + a32*(y2 - y1) + a33*(y3 - y1) + -(y1 + a12*(y2 - y1) +
  a13*(y3 - y1)) by A7,Th27
    .= y1 + a32*(y2 - y1) + a33*(y3 - y1) + (-y1 + -a12*(y2 - y1) + -a13*(y3
  - y1)) by Th8
    .= (y1 + -y1) + (a32*(y2 - y1) + -a12*(y2 - y1)) + (a33*(y3 - y1) + -a13
  *(y3 - y1)) by Th17
    .= 0*n + (a32*(y2 - y1) + -a12*(y2 - y1)) + (a33*(y3 - y1) + -a13*(y3 -
  y1)) by Th2
    .= 0*n + (a32 - a12)*(y2 - y1) + (a33*(y3 - y1) + -a13*(y3 - y1)) by Th11
    .= 0*n + (a32 - a12)*(y2 - y1) + (a33 - a13)*(y3 - y1) by Th11
    .= (a32 - a12)*(y2 - y1) + (a33 - a13)*(y3 - y1) by EUCLID_4:1;
A10: x1 = y1 + a12*(y2 - y1) + a13*(y3 - y1) by A7,Th27;
  then x2 - x1 = y1 + a22*(y2 - y1) + a23*(y3 - y1) + -(y1 + a12*(y2 - y1) +
  a13*(y3 - y1)) by A6,Th27
    .= y1 + a22*(y2 - y1) + a23*(y3 - y1) + (-y1 + -a12*(y2 - y1) + -a13*(y3
  - y1)) by Th8
    .= (y1 + -y1) + (a22*(y2 - y1) + -a12*(y2 - y1)) + (a23*(y3 - y1) + -a13
  *(y3 - y1)) by Th17
    .= 0*n + (a22*(y2 - y1) + -a12*(y2 - y1)) + (a23*(y3 - y1) + -a13*(y3 -
  y1)) by Th2
    .= 0*n + (a22 - a12)*(y2 - y1) + (a23*(y3 - y1) + -a13*(y3 - y1)) by Th11
    .= 0*n + (a22 - a12)*(y2 - y1) + (a23 - a13)*(y3 - y1) by Th11
    .= (a22 - a12)*(y2 - y1) + (a23 - a13)*(y3 - y1) by EUCLID_4:1;
  then consider c1,c2,d1,d2 be Real such that
A11: y2 - y1 = c1*(x2 - x1) + c2*(x3 - x1) & y3 - y1 = d1*(x2 - x1) + d2
  *(x3 - x1 ) by A4,A9,Th36;
A12: x1 = y1 + (a12*(y2 - y1) + a13*(y3 - y1)) by A10,RVSUM_1:15;
  now
    let y be object;
    assume y in P;
    then ex x9 be Element of REAL n st y = x9 &
   ex a19,a29,a39 being Real st
    a19+a29+a39=1 & x9 = a19*y1+a29*y2+a39 *y3 by A5;
    then consider b1,b2,b3 being Real such that
A13: b1+b2+b3=1 & y = b1*y1+b2*y2+b3*y3;
    y = y1 + b2*(y2 - y1) + b3*(y3 - y1) by A13,Th27
      .= x1 - (a12*(y2 - y1) + a13*(y3 - y1)) + b2*(y2 - y1) + b3*(y3 - y1)
    by A12,Th6
      .= x1 - a12*(y2 - y1) - a13*(y3 - y1) + b2*(y2 - y1) + b3*(y3 - y1) by
RVSUM_1:39
      .= x1 + -a12*(y2 - y1) + b2*(y2 - y1) + -a13*(y3 - y1) + b3*(y3 - y1)
    by RVSUM_1:15
      .= x1 + (-a12*(y2 - y1) + b2*(y2 - y1)) + -a13*(y3 - y1) + b3*(y3 - y1
    ) by RVSUM_1:15
      .= x1 + (-a12*(y2 - y1) + b2*(y2 - y1)) + (-a13*(y3 - y1) + b3*(y3 -
    y1)) by RVSUM_1:15
      .= x1 + (b2-a12)*(y2 - y1) + (-a13*(y3 - y1) + b3*(y3 - y1)) by Th11
      .= x1 + (b2-a12)*(y2 - y1) + (b3-a13)*(y3 - y1) by Th11
      .= x1 + ((b2-a12)*(c1*(x2 - x1)) + (b2-a12)*(c2*(x3 - x1))) + (b3-a13)
    *(d1*(x2 - x1) + d2*(x3 - x1)) by A11,EUCLID_4:6
      .= x1 + ((b2-a12)*(c1*(x2 - x1)) + (b2-a12)*(c2*(x3 - x1))) + ((b3-a13
    )*(d1*(x2 - x1)) + (b3-a13)*(d2*(x3 - x1))) by EUCLID_4:6
      .= x1 + ((((b2-a12)*c1)*(x2 - x1)) + (b2-a12)*(c2*(x3 - x1))) + ((b3-
    a13)*(d1*(x2 - x1)) + (b3-a13)*(d2*(x3 - x1))) by EUCLID_4:4
      .= x1 + (((b2-a12)*c1)*(x2 - x1) + ((b2-a12)*c2)*(x3 - x1)) + ((b3-a13
    )*(d1*(x2 - x1)) + (b3-a13)*(d2*(x3 - x1))) by EUCLID_4:4
      .= x1 + (((b2-a12)*c1)*(x2 - x1) + ((b2-a12)*c2)*(x3 - x1)) + (((b3-
    a13)*d1)*(x2 - x1) + (b3-a13)*(d2*(x3 - x1))) by EUCLID_4:4
      .= x1 + (((b2-a12)*c1)*(x2 - x1) + ((b2-a12)*c2)*(x3 - x1)) + (((b3-
    a13)*d1)*(x2 - x1) + ((b3-a13)*d2)*(x3 - x1)) by EUCLID_4:4
      .= x1 + ((b2-a12)*c1)*(x2 - x1) + ((b2-a12)*c2)*(x3 - x1) + (((b3-a13)
    *d1)*(x2 - x1) + ((b3-a13)*d2)*(x3 - x1)) by RVSUM_1:15
      .= x1 + ((b2-a12)*c1)*(x2 - x1) + ((b2-a12)*c2)*(x3 - x1) + ((b3-a13)*
    d1)*(x2 - x1) + ((b3-a13)*d2)*(x3 - x1) by RVSUM_1:15
      .= x1 + ((b2-a12)*c1)*(x2 - x1) + ((b3-a13)*d1)*(x2 - x1) + ((b2-a12)*
    c2)*(x3 - x1) + ((b3-a13)*d2)*(x3 - x1) by RVSUM_1:15
      .= x1 + (((b2-a12)*c1)*(x2 - x1) + ((b3-a13)*d1)*(x2 - x1)) + ((b2-a12
    )*c2)*(x3 - x1) + ((b3-a13)*d2)*(x3 - x1) by RVSUM_1:15
      .= x1 + (((b2-a12)*c1)*(x2 - x1) + ((b3-a13)*d1)*(x2 - x1)) + (((b2-
    a12)*c2)*(x3 - x1) + ((b3-a13)*d2)*(x3 - x1)) by RVSUM_1:15
      .= x1 + ((b2-a12)*c1 + (b3-a13)*d1)*(x2 - x1) + (((b2-a12)*c2)*(x3 -
    x1) + ((b3-a13)*d2)*(x3 - x1)) by EUCLID_4:7
      .= x1 + ((b2-a12)*c1 + (b3-a13)*d1)*(x2 - x1) + ((b2-a12)*c2 + (b3-a13
    )*d2)*(x3 - x1) by EUCLID_4:7;
    then ex a be Real st y = a*x1 + ((b2-a12)*c1 + (b3-a13)*d1)*x2 +((b2-a12)*
c2 + (b3-a13)*d2)* x3 & a+((b2-a12)*c1 + (b3-a13)*d1)+((b2-a12)*c2 + (b3-a13)*
    d2)=1 by Th28;
    hence y in plane(x1,x2,x3);
  end;
  then
A14: P c= plane(x1,x2,x3);
  plane(x1,x2,x3) c= P by A1,A2,A3,A5,Th83;
  hence thesis by A14,XBOOLE_0:def 10;
end;
