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;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);

theorem
  for POS being non empty ParOrtStr holds POS is OrtAfPl-like iff (ex a,
  b being Element of POS st a<>b) & for a,b,c,d,p,q,r,s being Element of POS
holds a,b // b,a & a,b // c,c & (a,b // p,q & a,b // r,s implies p,q // r,s or
a=b) & (a,b // a,c implies b,a // b,c) & (ex x being Element of POS st a,b // c
  ,x & a,c // b,x) & (ex x,y,z being Element of POS st not x,y // x,z ) & (ex x
  being Element of POS st a,b // c,x & c <>x) & (a,b // b,d & b<>a implies ex x
being Element of POS st c,b // b,x & c,a // d,x) & (a,b _|_ a,b implies a=b) &
a,b _|_ c,c & (a,b _|_ c,d implies a,b _|_ d,c & c,d _|_ a,b) & (a,b _|_ p,q &
a,b // r,s implies p,q _|_ r,s or a=b) & (a,b _|_ p,q & a,b _|_ r,s implies p,q
// r,s or a=b) & (ex x being Element of POS st a,b _|_ c,x & c <>x) & (not a,b
  // c,d implies ex x being Element of POS st a,b // a,x & c,d // c,x )
proof
  let POS be non empty ParOrtStr;
  set P = the AffinStruct of POS;
  hereby
    assume
A1: POS is OrtAfPl-like;
    then P is AffinPlane;
    hence ex x,y being Element of POS st x<>y by DIRAF:46;
    let a,b,c,d,p,q,r,s be Element of POS;
    reconsider a9=a,b9=b,c9=c,d9=d,p9=p,q9=q,r9=r,s9=s as Element of P;
    consider x9 being Element of P such that
A2: a9,b9 // c9,x9 & a9,c9 // b9,x9 by A1,DIRAF:46;
    a9,b9 // b9,a9 & a9,b9 // c9,c9 by A1,DIRAF:46;
    hence a,b // b,a & a,b // c,c by Th36;
    hereby
      assume a,b // p,q & a,b // r,s;
      then a9,b9 // p9,q9 & a9,b9 // r9,s9 by Th36;
      then p9,q9 // r9,s9 or a9=b9 by A1,DIRAF:46;
      hence p,q // r,s or a=b by Th36;
    end;
    hereby
      assume a,b // a,c;
      then a9,b9 // a9,c9 by Th36;
      then b9,a9 // b9,c9 by A1,DIRAF:46;
      hence b,a // b,c by Th36;
    end;
    reconsider x=x9 as Element of POS;
    consider x9,y9,z9 being Element of P such that
A3: not x9,y9 // x9,z9 by A1,DIRAF:46;
    a,b // c,x & a,c // b,x by A2,Th36;
    hence ex x being Element of POS st a,b // c,x & a,c // b,x;
    reconsider x=x9,y=y9,z=z9 as Element of POS;
    consider x9 being Element of P such that
A4: a9,b9 // c9,x9 and
A5: c9<>x9 by A1,DIRAF:46;
    not x,y // x,z by A3,Th36;
    hence ex x,y,z being Element of POS st not x,y // x,z;
    reconsider x=x9 as Element of POS;
    a,b // c,x by A4,Th36;
    hence ex x being Element of POS st a,b // c,x & c <>x by A5;
    hereby
      assume that
A6:   a,b // b,d and
A7:   b<>a;
      a9,b9 // b9,d9 by A6,Th36;
      then consider x9 being Element of P such that
A8:   c9,b9 // b9,x9 & c9,a9 // d9,x9 by A1,A7,DIRAF:46;
      reconsider x=x9 as Element of POS;
      c,b // b,x & c,a // d,x by A8,Th36;
      hence ex x being Element of POS st c,b // b,x & c,a // d,x;
    end;
    thus (a,b _|_ a,b implies a=b) & a,b _|_ c,c & (a,b _|_ c,d implies a,b
_|_ d,c & c,d _|_ a,b) & (a,b _|_ p,q & a,b // r,s implies p,q _|_ r,s or a=b)
& (a,b _|_ p,q & a,b _|_ r,s implies p,q // r,s or a=b) & ex x being Element of
    POS st a,b _|_ c,x & c <>x by A1;
    assume not a,b // c,d;
    then not a9,b9 // c9,d9 by Th36;
    then consider x9 being Element of P such that
A9: a9,b9 // a9,x9 & c9,d9 // c9,x9 by A1,DIRAF:46;
    reconsider x=x9 as Element of POS;
    a,b // a,x & c,d // c,x by A9,Th36;
    hence ex x being Element of POS st a,b // a,x & c,d // c,x;
  end;
  given a,b being Element of POS such that
A10: a<>b;
  assume
A11: for a,b,c,d,p,q,r,s being Element of POS holds a,b // b,a & a,b //
c,c & (a,b // p,q & a,b // r,s implies p,q // r,s or a=b) & (a,b // a,c implies
b,a // b,c) & (ex x being Element of POS st a,b // c,x & a,c // b,x) & (ex x,y,
z being Element of POS st not x,y // x,z ) & (ex x being Element of POS st a,b
  // c,x & c <>x) & (a,b // b,d & b<>a implies ex x being Element of POS st c,b
  // b,x & c,a // d,x) & (a,b _|_ a,b implies a=b) & a,b _|_ c,c & (a,b _|_ c,d
implies a,b _|_ d,c & c,d _|_ a,b) & (a,b _|_ p,q & a,b // r,s implies p,q _|_
  r,s or a=b) & (a,b _|_ p,q & a,b _|_ r,s implies p,q // r,s or a=b) & (ex x
  being Element of POS st a,b _|_ c,x & c <>x) & (not a,b // c,d implies ex x
  being Element of POS st a,b // a,x & c,d // c,x );
A12: now
    let x,y,z be Element of P;
    reconsider x9=x,y9=y,z9=z as Element of POS;
    consider t9 being Element of POS such that
A13: x9,z9 // y9,t9 and
A14: y9<>t9 by A11;
    reconsider t=t9 as Element of P;
    x,z // y,t by A13,Th36;
    hence ex t being Element of P st x,z // y,t & y<>t by A14;
  end;
A15: now
    let x,y,z,t,u,w be Element of P;
    reconsider a=x,b=y,c =z,d=t,e=u,f=w as Element of POS;
    a,b // b,a & a,b // c,c by A11;
    hence x,y // y,x & x,y // z,z by Th36;
    thus x<>y & x,y // z,t & x,y // u,w implies z,t // u,w
    proof
      assume that
A16:  x<>y and
A17:  x,y // z,t & x,y // u,w;
      a,b // c,d & a,b // e,f by A17,Th36;
      then c,d // e,f by A11,A16;
      hence thesis by Th36;
    end;
    thus x,y // x,z implies y,x // y,z
    proof
      assume x,y // x,z;
      then a,b // a, c by Th36;
      then b,a // b,c by A11;
      hence thesis by Th36;
    end;
  end;
A18: now
    let x,y,z,t be Element of P such that
A19: not x,y // z,t;
    reconsider x9=x,y9=y,z9=z,t9=t as Element of POS;
    not x9,y9 // z9,t9 by A19,Th36;
    then consider u9 being Element of POS such that
A20: x9,y9 // x9,u9 & z9,t9 // z9,u9 by A11;
    reconsider u=u9 as Element of P;
    x,y // x,u & z,t // z,u by A20,Th36;
    hence ex u being Element of P st x,y // x,u & z,t // z,u;
  end;
A21: now
    let x,y,z,t be Element of P such that
A22: z,x // x,t and
A23: x<>z;
    reconsider x9=x,y9=y,z9=z,t9=t as Element of POS;
    z9,x9 // x9,t9 by A22,Th36;
    then consider u9 being Element of POS such that
A24: y9,x9 // x9,u9 & y9,z9 // t9,u9 by A11,A23;
    reconsider u=u9 as Element of P;
    y,x // x,u & y,z // t,u by A24,Th36;
    hence ex u being Element of P st y,x // x,u & y,z // t,u;
  end;
A25: now
    let x,y,z be Element of P;
    reconsider x9=x,y9=y,z9=z as Element of POS;
    consider t9 being Element of POS such that
A26: x9,y9 // z9,t9 & x9,z9 // y9,t9 by A11;
    reconsider t=t9 as Element of P;
    x,y // z,t & x,z // y,t by A26,Th36;
    hence ex t being Element of P st x,y // z,t & x,z // y,t;
  end;
  ex x,y,z being Element of P st not x,y // x,z
  proof
    consider x,y,z being Element of POS such that
A27: not x,y // x,z by A11;
    reconsider x9=x,y9=y,z9=z as Element of P;
    not x9,y9 // x9,z9 by A27,Th36;
    hence thesis;
  end;
  hence
  AffinStruct(#the carrier of POS,the CONGR of POS#) is AffinPlane by A10,A15
,A12,A25,A21,A18,DIRAF:46;
  thus for a,b,c,d,p,q,r,s be Element of POS holds (a,b _|_ a,b implies a=b) &
a,b _|_ c,c & (a,b _|_ c,d implies a,b _|_ d,c & c,d _|_ a,b) & (a,b _|_ p,q &
a,b // r,s implies p,q _|_ r,s or a=b) & (a,b _|_ p,q & a,b _|_ r,s implies p,q
  // r,s or a=b) by A11;
  thus thesis by A11;
end;
