reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th25:
  transl(p1,TOP-REAL n) .: Plane(p,p2) = Plane(p,p1+p2)
proof
A1: now
    let y be object;
    assume y in transl(p1,TOP-REAL n) .: Plane(p,p2);
    then consider x be object such that
A2: [x,y] in transl(p1,TOP-REAL n) & x in Plane(p,p2) by RELAT_1:def 13;
    consider x1 be Point of TOP-REAL n such that
A3: x = x1 & |( p,x1-p2 )| = 0 by A2;
A4: y = transl(p1,TOP-REAL n).x1 by A2,A3,FUNCT_1:1
    .= p1+x1 by RLTOPSP1:def 10;
    x in dom transl(p1,TOP-REAL n) &
    y = transl(p1,TOP-REAL n).x by A2,FUNCT_1:1;
    then y in rng transl(p1,TOP-REAL n) by FUNCT_1:3;
    then reconsider y1 = y as Point of TOP-REAL n;
    x1-p2 = x1-p2+0.TOP-REAL n by RLVECT_1:4
    .= x1-p2+(p1 + -p1) by RLVECT_1:5
    .= x1 + -p2 +p1 + -p1 by RLVECT_1:def 3
    .= y1 -p2 -p1 by A4,RLVECT_1:def 3
    .= y1 -(p1+p2) by RLVECT_1:27;
    hence y in Plane(p,p1+p2) by A3;
  end;
  now
    let y be object;
    assume y in Plane(p,p1+p2);
    then consider y1 be Point of TOP-REAL n such that
A5: y = y1 & |( p,y1-(p1+p2) )| = 0;
    set x = y1-p1;
    x in the carrier of TOP-REAL n; then
A6: x in dom transl(p1,TOP-REAL n) by FUNCT_2:def 1;
    p1 + x = y1 by RLVECT_4:1;
    then transl(p1,TOP-REAL n).x = y1 by RLTOPSP1:def 10; then
A7: [x,y1] in transl(p1,TOP-REAL n) by A6,FUNCT_1:1;
    x-p2 = y1 -(p1+p2) by RLVECT_1:27;
    then x in Plane(p,p2) by A5;
    hence y in transl(p1,TOP-REAL n) .: Plane(p,p2)
    by A5,A7,RELAT_1:def 13;
  end;
  hence thesis by A1,TARSKI:2;
end;
