reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;
reserve t,s,r1 for Real;
reserve n for Element of NAT;
reserve X,Y,B1,B2 for Subset of TOP-REAL n;
reserve x,y for Point of TOP-REAL n;

theorem Th62:
  t(.)(X+x) = t(.)X+t*x
proof
  thus t(.)(X+x) c= t(.)X+t*x
  proof
    let b be object;
    assume b in t(.)(X+x);
    then consider a being Point of TOP-REAL n such that
A1: b = t*a and
A2: a in X+x;
    consider x1 being Point of TOP-REAL n such that
A3: a=x1+x and
A4: x1 in X by A2;
A5: t*x1 in t(.)X by A4;
    b=t*x1+t*x by A1,A3,RLVECT_1:def 5;
    hence thesis by A5;
  end;
  let b be object;
  assume b in t(.)X+t*x;
  then consider x1 being Point of TOP-REAL n such that
A6: b = x1+t*x and
A7: x1 in t(.)X;
  consider a being Point of TOP-REAL n such that
A8: x1=t*a and
A9: a in X by A7;
A10: a+x in X+x by A9;
  b=t*(a+x) by A6,A8,RLVECT_1:def 5;
  hence thesis by A10;
end;
