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 Th83:
  f is translation & g is translation & f.a=g.a & not LIN a,f.a,x
  implies f.x=g.x
proof
  assume that
A1: f is translation and
A2: g is translation and
A3: f.a=g.a and
A4: not LIN a,f.a,x;
  set b=f.a, y=f.x, z=g.x;
A5: a,x // b,z & a,b // x,z by A2,A3,Th68,Th82;
 f is dilatation by A1;
  then
A6: a,x // b,y by Th68;
  a,b // x,y by A1,Th82;
  hence thesis by A4,A6,A5,Th80;
end;
