reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];
reserve AS for non empty AffinStruct;
reserve a,b,x,y for Element of AS;
reserve CS for CongrSpace;
reserve OAS for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u for Element of OAS;
reserve f,g for Permutation of the carrier of OAS;
reserve AFS for AffinSpace;
reserve a,b,c,d,d1,d2,p,x,y,z,t for Element of AFS;
reserve f,g for Permutation of the carrier of AFS;

theorem Th80:
  not LIN a,b,c & a,b // c,d1 & a,b // c,d2 & a,c // b,d1 & a,c //
  b,d2 implies d1=d2
proof
  assume that
A1: not LIN a,b,c and
A2: a,b // c,d1 and
A3: a,b // c,d2 and
A4: a,c // b,d1 and
A5: a,c // b,d2;
  assume
A6: d1<>d2;
  a<>c by A1,AFF_1:7;
  then b,d1 // b,d2 by A4,A5,AFF_1:5;
  then LIN b,d1,d2 by AFF_1:def 1;
  then
A7: LIN d1,d2,b by AFF_1:6;
A8: now
    assume c = d1;
    then c,a // c,b by A4,AFF_1:4;
    then LIN c,a,b by AFF_1:def 1;
    hence contradiction by A1,AFF_1:6;
  end;
A9: LIN d1,d2,d1 by AFF_1:7;
  a<>b by A1,AFF_1:7;
  then c,d1 // c,d2 by A2,A3,AFF_1:5;
  then
A10: LIN c,d1,d2 by AFF_1:def 1;
  then
A11: LIN d1,d2,c by AFF_1:6;
  LIN d1,d2,c by A10,AFF_1:6;
  then d1,d2 // c,d1 by A9,AFF_1:10;
  then d1,d2 // a,b or c = d1 by A2,AFF_1:5;
  then d1,d2 // b,a by A8,AFF_1:4;
  then LIN d1,d2,a by A6,A7,AFF_1:9;
  hence contradiction by A1,A6,A11,A7,AFF_1:8;
end;
