
theorem lem4:
for f being ascending Field-yielding sequence
for a,b being Element of SeqField f
for i being Element of NAT, x,y being Element of f.i
st x = a & y = b holds a + b = x + y & a * b = x * y
proof
let f be ascending Field-yielding sequence, a,b be Element of SeqField f;
let i be Element of NAT, x,y be Element of f.i;
assume AS: x = a & y = b;
H: a is Element of Carrier f & b is Element of Carrier f by dsf; then
consider j being Element of NAT, xj,yj being Element of f.j such that
A: xj = a & yj = b & (addseq f).(a,b) = xj + yj by defadd;
x + y = xj + yj by AS,A,lem2;
hence a + b = x + y by A,dsf;
consider j being Element of NAT, xj,yj being Element of f.j such that
A: xj = a & yj = b & (multseq f).(a,b) = xj * yj by H,defmult;
x * y = xj * yj by AS,A,lem2;
hence a * b = x * y by A,dsf;
end;
