reserve AS for AffinSpace,
  a,b,c,d,p,q,r,s,x for Element of AS;
reserve AFP for AffinPlane,
  a,a9,b,b9,c,c9,d,p,p9,q,q9,r,x,x9,y,y9,z for Element of AFP,
  A,C,P for Subset of AFP,
  f,g,h,f1,f2 for Permutation of the carrier of AFP;

theorem
  AFP is translational & AFP is Fanoian & f1 is translation & f2 is
  translation & g=f1*f1 & g=f2*f2 implies f1=f2
proof
  assume that
A1: AFP is translational and
A2: AFP is Fanoian and
A3: f1 is translation and
A4: f2 is translation and
A5: g=f1*f1 & g=f2*f2;
  set h=f2"*f1;
A6: f2" is translation by A4,TRANSGEO:85;
  h*h=f2"*(f1*(f2"*f1)) by RELAT_1:36
    .=f2"*((f1*f2")*f1) by RELAT_1:36
    .=f2"*((f2"*f1)*f1) by A1,A3,A6,Th10
    .=f2"*(f2"*(f1*f1)) by RELAT_1:36
    .=(f2"*f2")*(f1*f1) by RELAT_1:36
    .=g"*g by A5,FUNCT_1:44
    .=id the carrier of AFP by FUNCT_2:61;
  then f2"*f1=id the carrier of AFP by A2,A3,A6,Th13,TRANSGEO:86;
  then f2"*f1=f2"*f2 by FUNCT_2:61;
  hence thesis by TRANSGEO:5;
end;
