theorem Th43:
  f,g equal_at m,n & g,h equal_at m,n implies f,h equal_at m,n
proof
  assume that
A1: f,g equal_at m,n and
A2: g,h equal_at m,n;
A3: dom f = dom g by A1;
A4: now
    let k;
    assume
A5: k in dom f & m <=k & k <= n;
    hence f.k=g.k by A1
      .=h.k by A2,A3,A5;
  end;
  dom g = dom h by A2;
  hence thesis by A3,A4;
end;
