reserve i,j,k,m,n for Nat;
reserve R for non empty doubleLoopStr;
reserve x,y for Scalar of R;
reserve f,g,h for FinSequence of R;

theorem
  x is being_a_product_of_squares & y is being_a_square implies x*y is
  being_a_product_of_squares
proof
  assume that
A1: x is being_a_product_of_squares and
A2: y is being_a_square;
  consider f such that
A3: f is being_a_Product_of_squares & x=f/.len f by A1;
  take g=f^<*x*y*>;
  len g=len f+len <*x*y*> by FINSEQ_1:22
    .=len f+1 by Lm2;
  hence thesis by A2,A3,Lm3,Lm83;
end;
