
theorem Th19f:
for E being Field,
    F being Subfield of E
for x being non zero Element of E,
    x1 being Element of F st x = x1 holds x" = x1"
proof
let R be Field, S being Subfield of R;
let x be non zero Element of R, x1 be Element of S;
set C = the carrier of R, C1 = the carrier of S, a = x1";
assume A1: x = x1;
A2: x <> 0.R; then
A3: x1 <> 0.S by A1,EC_PF_1:def 1;
C1 c= C by EC_PF_1:def 1;
then reconsider g = a as Element of R;
R is FieldExtension of S by FIELD_4:7; then
S is Subring of R by FIELD_4:def 1; then
g * x = a * x1 by A1,Th18 .= 1.S by A3,VECTSP_1:def 10 .= 1.R by EC_PF_1:def 1;
hence thesis by A2,VECTSP_1:def 10;
end;
