let n be Element of NAT ; :: thesis: for z being Element of F_Complex st z is Real holds
ex x being Real st
( x = z & . (z,n) = x |^ n )

let z be Element of F_Complex; :: thesis: ( z is Real implies ex x being Real st
( x = z & . (z,n) = x |^ n ) )

assume z is Real ; :: thesis: ex x being Real st
( x = z & . (z,n) = x |^ n )

then reconsider x = z as Real ;
per cases ( x = 0 or x <> 0 ) ;
suppose A1: x = 0 ; :: thesis: ex x being Real st
( x = z & . (z,n) = x |^ n )

then A2: z = 0. F_Complex by COMPLFLD:def 1;
thus ex x being Real st
( x = z & . (z,n) = x |^ n ) :: thesis: verum
proof
per cases ( n = 0 or n > 0 ) ;
suppose A3: n = 0 ; :: thesis: ex x being Real st
( x = z & . (z,n) = x |^ n )

then . (z,n) = 1 by
.= x |^ n by ;
hence ex x being Real st
( x = z & . (z,n) = x |^ n ) ; :: thesis: verum
end;
suppose A4: n > 0 ; :: thesis: ex x being Real st
( x = z & . (z,n) = x |^ n )

then A5: n >= 0 + 1 by NAT_1:13;
. (z,n) = 0. F_Complex by
.= x |^ n by ;
hence ex x being Real st
( x = z & . (z,n) = x |^ n ) ; :: thesis: verum
end;
end;
end;
end;
suppose A6: x <> 0 ; :: thesis: ex x being Real st
( x = z & . (z,n) = x |^ n )

defpred S1[ Nat] means . (z,\$1) = x |^ \$1;
A7: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; :: thesis: ( S1[n] implies S1[n + 1] )
assume A8: S1[n] ; :: thesis: S1[n + 1]
reconsider nn = n as Element of NAT by ORDINAL1:def 12;
. (z,(n + 1)) = ( . (z,nn)) * z by GROUP_1:def 7
.= (x #Z n) * x by
.= (x #Z n) * (x #Z 1) by PREPOWER:35
.= x #Z (n + 1) by
.= x |^ (n + 1) by PREPOWER:36 ;
hence S1[n + 1] ; :: thesis: verum
end;
. (z,0) = 1r by
.= x #Z 0 by PREPOWER:34
.= x |^ 0 by PREPOWER:36 ;
then A9: S1[ 0 ] ;
for n being Nat holds S1[n] from NAT_1:sch 2(A9, A7);
then (power F_Complex) . (z,n) = x |^ n ;
hence ex x being Real st
( x = z & . (z,n) = x |^ n ) ; :: thesis: verum
end;
end;