reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem Th14:
  r*(seq1 - seq2) = r*seq1 - r*seq2
proof
  thus r*(seq1-seq2)=r*seq1+r*(-seq2) by Th12
    .=r*seq1+r*((-1)*seq2) by Th11
    .=r*seq1+((-1)*r)*seq2 by Th13
    .=r*seq1+(-1)*(r*seq2) by Th13
    .=r*seq1-(r*seq2) by Th11;
end;
